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Vasectomy
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Vasectomy
Background
Type
Sterilization
First use
1899 (experiments from 1785)[1]
Failure rates (first year)
Perfect use
0.10%[2]
Typical use
0.15%[2]
"Vas-Clip" nearly 1%
Usage
Duration effect
Permanent
Reversibility
Possible, but expensive.
User reminders
Two consecutive negative semen specimens required to verify no sperm.
Advantages and disadvantages
STI protection
No
Benefits
No need for general anesthesia. Lower cost and less invasive than tubal ligation for women.
Risks
Temporary local inflammation of the testes, long-term genital pain.
Vasectomy is a surgical procedure for male sterilization or permanent contraception. During the procedure, the male vas deferens are cut and tied or sealed so as to prevent sperm from entering into the urethra and thereby prevent fertilization. Vasectomies are usually performed in a physician's office, medical clinic, or, when performed on an animal, in a veterinary clinic—hospitalization is not normally required as the procedure is not complicated, the incisions are small, and the necessary equipment routine.
There are several methods by which a surgeon might complete a vasectomy procedure, all of which occlude (i.e., "seal") at least one side of each vas deferens. To help reduce anxiety and increase patient comfort, men who have an aversion to needles may consider a "no-needle" application of anesthesia while the "no-scalpel" or "open-ended" techniques help to accelerate recovery times and increase the chance of healthy recovery.
Due to the simplicity of the surgery, a vasectomy usually takes less than thirty minutes to complete. After a short recovery at the doctor's office (usually less than an hour), the patient is sent home to rest. Because the procedure is minimally invasive, many vasectomy patients find that they can resume their typical sexual behavior within a week, and do so with little or no discomfort.
Because the procedure is considered a permanent method of contraception and is not easily reversed, men are usually counseled/advised to consider how the long-term outcome of a vasectomy might affect them both emotionally and physically. The procedure is not often encouraged for young single childless men as their chances for biological parenthood are thereby more or less permanently reduced to almost zero. It is seldom performed on dogs (castration, a different procedure, remains the preferred reproductive control option for canines) but is regularly performed on bulls.[3]
Contents
[hide]
1 Medical uses
1.1 Efficacy
2 Complications
2.1 Postvasectomy pain
2.2 Psychological effects
2.3 Dementia
3 Procedure
3.1 Other techniques
3.1.1 Vas occlusion techniques
3.2 Recovery
4 Conceiving after vasectomy
4.1 Vasectomy reversal
5 Prevalence
6 History
7 Society and culture
7.1 Availability and legality
7.2 Ideological issues
7.3 Tourism
8 See also
9 References
10 External links
Medical uses[edit]
A vasectomy is done to prevent fertility in males. It ensures that in most cases the person will be sterile after confirmation of success following surgery. The procedure is regarded as permanent because vasectomy reversal is costly and often does not restore the male's sperm count or sperm motility to prevasectomy levels. Men with vasectomies have a very small (nearly zero) chance of successfully impregnating a woman, but a vasectomy has no effect on rates of sexually transmitted infections.
After vasectomy, the testes remain in the scrotum where Leydig cells continue to produce testosterone and other male hormones that continue to be secreted into the blood-stream. Some studies have found that sexual desire after vasectomy may be somewhat diminished.[4][5]
When the vasectomy is complete, sperm cannot exit the body through the penis. Sperm are still produced by the testicles, but they are soon broken down and absorbed by the body. Much fluid content is absorbed by membranes in the epididymis, and much solid content is broken down by the responding macrophages and reabsorbed via the blood stream. Sperm is matured in the epididymis for about a month before leaving the testicles. After vasectomy, the membranes must increase in size to absorb and store more fluid; this triggering of the immune system causes more macrophages to be recruited to break down and reabsorb more solid content. Within one year after vasectomy, sixty to seventy per cent of vasectomized men develop antisperm antibodies.[6] In some cases, vasitis nodosa, a benign proliferation of the ductular epithelium, can also result.[7][8] The accumulation of sperm increases pressure in the vas deferens and epididymis. The entry of the sperm into the scrotum can cause sperm granulomas to be formed by the body to contain and absorb the sperm which the body will treat as a foreign biological substance (much like a virus or bacterium).[9]
Efficacy[edit]
Vasectomy is the most effective permanent form of contraception available to men. In nearly every way that vasectomy can be compared to tubal ligation it has a more positive outlook. Vasectomy is more cost effective, less invasive, has techniques that are emerging that may facilitate easier reversal, and has a much lower risk of postoperative complications. Early failure rates, i.e. pregnancy within a few months after vasectomy, typically result from unprotected sexual intercourse too soon after the procedure while some sperm continue to pass through the vasa deferentia. Most physicians and surgeons who perform vasectomies recommend one (sometimes two) postprocedural semen specimens to verify a successful vasectomy; however, many men fail to return for verification tests citing inconvenience, embarrassment, forgetfulness, or certainty of sterility.[10] In January 2008, the F.D.A. cleared a home test called SpermCheck Vasectomy that allows patients to perform postvasectomy confirmation tests themselves;[11] however, compliance for postvasectomy semen analysis in general remains low.
Late failure, i.e. pregnancy following spontaneous recanalization of the vasa deferentia, has also been documented.[12] The Royal College of Obstetricians and Gynaecologists states there is a generally agreed-upon rate of late failure of about one in 2000 vasectomies— better than tubal ligations for which the failure rate is one in every 200 to 300 cases.[13] A 2005 review including both early and late failures described a total of 183 recanalizations from 43,642 vasectomies (0.4%), and sixty pregnancies after 92,184 vasectomies (0.07%).[14]
Complications[edit]
Short-term possible complications include infection, bruising and bleeding into the scrotum resulting in a collection of blood known as a hematoma. A study in 2012 demonstrated an infection rate of 2.5% postvasectomy.[15] The stitches on the small incisions required are prone to irritation, though this can be minimized by covering them with gauze or small adhesive bandages. The primary long-term complications are chronic pain conditions or syndromes that can affect any of the scrotal, pelvic or lower-abdominal regions, collectively known as post-vasectomy pain syndrome. Though vasectomy results in increases in circulating immune complexes, these increases are transient. Data based on animal and human studies indicate these changes do not result in increased incidence of atherosclerosis. The risk of prostate and testicular cancer is not affected by vasectomy.[16][17][18] According to one study long-term postvasectomy discomfort is experienced at a frequency which ranges between 15% and 33% of vasectomy patients.[19]
Postvasectomy pain[edit]
Main article: Post-vasectomy pain syndrome
Post-vasectomy pain syndrome (P.V.P.S.) is a chronic and sometimes debilitating condition that may develop immediately or several years after vasectomy.[20] One survey cites studies that estimate incidence at one case in every ten to thirty vasectomies.[21] The pain can be constant orchialgia or epididymal pain (epididymitis), or it can be pain that occurs only at particular times such as with sexual intercourse, ejaculation, or physical exertion.[9]
Psychological effects[edit]
Approximately 90% are generally reported in reviews as being satisfied with having had a vasectomy,[22] while 7–10% of men regret their decision.[23] Regret was less common when both people in the relationship agreed on the procedure.[24]
Men who are of a younger age at the time of having a vasectomy are significantly more likely to regret and seek a reversal of their vasectomy, with one study showing men for example in their twenties being 12.5 times more likely to undergo a vasectomy reversal later in life (and including some who chose sterilization at a young age);[25] prevasectomy counseling is therefore of particular importance in younger patients.
Dementia[edit]
An association between vasectomy and primary progressive aphasia, a rare variety of frontotemporal dementia, was reported.[24] However, it is unclear if there is a causal relationship.[26] The putative mechanism is a cross-reactivity between brain and sperm, including the shared presence of neural surface antigens.[27] In addition, the cytoskeletal tau protein has been found only to exist outside of the CNS in the manchette of sperm.[27]
Procedure[edit]
The traditional incision approach of vasectomy involves numbing of the scrotum with local anesthetic (although some men's physiology may make access to the vas deferens more difficult in which case general anesthesia may be recommended) after which a scalpel is used to make two small incisions, one on each side of the scrotum at a location that allows the surgeon to bring each vas deferens to the surface for excision. The vasa deferentia are cut (sometimes a section may be removed altogether), separated, and then at least one side is sealed by ligating (suturing), cauterizing (electrocauterization), or clamping.[28] There are several variations to this method that improve healing, effectiveness, and which help mitigate long-term pain such as post-vasectomy pain syndrome (PVPS) or epididymitis.
Fascial interposition: Recanalization of the vas deferens is a known cause of vasectomy failure(s).[29] Fascial interposition ("FI"), in which a tissue barrier is placed between the cut ends of the vas by suturing, may help to prevent this type of failure, increasing the overall success rate of vasectomy while leaving the testicular end within the confines of the fascia.[30] The fascia is a fibrous protective sheath that surrounds the vas deferens as well as all other body muscle tissue. This method, when combined with intraluminal cautery (where one or both sides of the vas deferens are electrically "burned" closed to prevent recanalization), has been shown to increase the success rate of vasectomy procedures.
No-needle anesthesia: Fear of needles for injection of local anesthesia is well known.[31] In 2005, a method of local anesthesia was introduced for vasectomy which allows the surgeon to apply it painlessly with a special jet-injection tool, as opposed to traditional needle application. The numbing agent is forced/pushed onto and deep enough into the scrotal tissue to allow for a virtually pain-free surgery. Initial surveys show a very high satisfaction rate amongst vasectomy patients.[31] Once the effects of no-needle anesthesia set in, the vasectomy procedure is performed in the routine manner.
No-scalpel vasectomy (NSV): Also known as a "key-hole" vasectomy,[28] is a vasectomy in which a sharp hemostat (as opposed to a scalpel) is used to puncture the scrotum. This method has come into widespread use as the resulting smaller "incision" or puncture wound typically limits bleeding and hematomas. Also the smaller wound has less chance of infection, resulting in faster healing times compared to the larger/longer incisions made with a scalpel. The surgical wound created by the No-Scalpel method usually does not require stitches. NSV is the most commonly performed type of minimally invasive vasectomy, and both describe the method of vasectomy that leads to access of the vas deferens.[32]
Open-ended vasectomy: In this procedure the testicular end of the vas deferens is not sealed, which allows continued streaming of sperm into the scrotum. This method may avoid testicular pain resulting from increased back-pressure in the epididymis.[9] Studies suggest that this method may reduce long-term complications such as post-vasectomy pain syndrome.[33][34]
Vas irrigation: Injections of sterile water or euflavine (which kills sperm) are put into the distal portion of the vas at the time of surgery which then brings about a near-immediate sterile ("azoospermatic") condition. The use of euflavine does however, tend to decrease time (or, number of) ejaculations to azoospermia vs. the water irrigation by itself. This additional step in the vasectomy procedure, (and similarly, fascial interposition), has shown positive results but is not as prominently in use, and few surgeons offer it as part of their vasectomy procedure.[35]
Other techniques[edit]
The following vasectomy methods have purportedly had a better chance of later reversal but have seen less use by virtue of known higher failure rates (i.e., recanalization). An earlier clip device, the VasClip, is no longer on the market, due to unacceptably high failure rates.[36][37][38]
The VasClip method, though considered reversible, has had a higher cost and resulted in lower success rates. Also, because the vasa deferentia are not cut or tied with this method, it could technically be classified as other than a vasectomy. Vasectomy reversal (and the success thereof) was conjectured to be higher as it only required removing the Vas-Clip device. This method achieved limited use, and scant reversal data are available.[38]
Vas occlusion techniques[edit]
Injected plugs: There are two types of injected plugs which can be used to block the vasa deferentia. Medical-grade polyurethane (M.P.U.) or medical-grade silicone rubber (MSR) starts as a liquid polymer that is injected into the vas deferens after which the liquid is clamped in place until is solidifies (usually in a few minutes).[39]
Intra-vas device: The vasa deferentia can also be occluded by an intra-vas device (IVD). A small cut is made in the lower abdomen after which a soft silicone or urethane plug is inserted into each vas tube thereby blocking (occluding) sperm. This method allows for the vas to remain intact. IVD technique is done in an out-patient setting with local anesthetic, similar to a traditional vasectomy. IVD reversal can be performed under the same conditions making it much less costly than vasovasostomy which can require general anesthesia and longer surgery time.[40]
Both vas occlusion techniques require the same basic patient setup: local anesthesia, puncturing of the scrotal sac for access of the vas, and then plug or injected plug occlusion. The success of the aforementioned vas occlusion techniques is not clear and data are still limited. Studies have shown, however, that the time to achieve sterility is longer than the more prominent techniques mentioned in the beginning of this article. The satisfaction rate of patients undergoing IVD techniques has a high rate of satisfaction with regard to the surgery experience itself.[35]
Recovery[edit]
A post-vasectomy scrotum, showing typical post-operative bruising, incision stitches and a shaved scrotum.
Sexual intercourse can usually be resumed in about a week (depending on recovery); however, pregnancy is still possible as long as the sperm count is above zero. Another method of contraception must be relied upon until a sperm count is performed either two months after the vasectomy or after ten to twenty ejaculations have occurred.[41]
After a vasectomy, contraceptive precautions must be continued until azoospermia is confirmed. Usually two semen analyses at three and four months are necessary to confirm azoospermia. The British Andrological Society has recommended that a single semen analysis confirming azoospermia after sixteen weeks is sufficient.[42]
Conceiving after vasectomy[edit]
In order to allow the possibility of reproduction via artificial insemination after vasectomy, some men opt for cryopreservation of sperm before sterilization. It is advised that all men having a vasectomy consider freezing some sperm before the procedure. Dr Allan Pacey (senior lecturer in andrology at Sheffield University and secretary of the British Fertility Society) notes that men who he sees for a vasectomy reversal which has not worked, express wishing they knew they could have stored sperm. Pacey notes, "The problem is you're asking a man to foresee a future where he might not necessarily be with his current partner—and that may be quite hard to do when she's sitting next to you."[43]
The cost of cryopreservation (sperm banking) itself may also be substantially less (approximately every five years) than alternative vaso-vasectomy procedure(s), compared to the costs of in-vitro fertilization which usually run to US$15,000.[citation needed]
Intracytoplasmic sperm injection: Sperm can be aspirated from the testicles or the epididymis, and while there is not enough for successful artificial insemination, there is enough to fertilize an ovum by ICSI This avoids the problem of antisperm antibodies and may result in a faster pregnancy. IVF (In-vitro fertilization) may be less costly (see aforementioned text) per cycle than reversal in some health-care systems, but a single I.V.F. cycle is often insufficient for conception. Disadvantages include the need for procedures on the woman, and the standard potential side-effects of I.V.F. for both the mother and the child.[44]
Vasectomy reversal[edit]
Main article: Vasectomy reversal
Although men considering vasectomies should not think of them as reversible, and most men and their partners are satisfied with the operation,[45][46] life circumstances and outlooks can change, and there is a surgical procedure to reverse vasectomies using vasovasostomy (a form of microsurgery first performed by Earl Owen in 1971[47][48]). Vasovasostomy is effective at achieving pregnancy in a variable percentage of cases, and total out-of-pocket costs in the United States are often upwards of $10,000.[49] The typical success rate of pregnancy following a vasectomy reversal is around 55% if performed within 10 years, and drops to around 25% if performed after 10 years.[50] After reversal, sperm counts and motility are usually much lower than pre-vasectomy levels. There is evidence that men who have had a vasectomy may produce more abnormal sperm, which would explain why even a mechanically successful reversal does not always restore fertility.[51][52] The higher rates of aneuploidy and diploidy in the sperm cells of men who have undergone vasectomy reversal may lead to a higher rate of birth defects.[51]
Some reasons that men seek vasectomy reversals include wanting a family with a new partner following a relationship break-down/divorce, their original wife/partner dying and subsequently going on to repartner and to want children, the unexpected death of a child, or a long-standing couple changing their minds some time later, often prompted by situations such as improved finances or existing children approaching the age of school or leaving home.[43] Patients often comment that they never anticipated the possibility of a relationship break-down or death (of their partner or child), or how that may affect their situation at the time of having their vasectomy. A small number of vasectomy reversals are also performed in attempts to relieve postvasectomy pain syndrome.[53]
Prevalence[edit]
Internationally, vasectomy rates are vastly different.[54] While female sterilisation is the most widely used method worldwide, with 223 million women relying on it, only 28 million women rely on their partner’s vasectomy.[55] In the world’s 69 least developed countries less than 0.1% of men use vasectomies on average. Of 54 African countries, only ten report measurable vasectomy use and only Swaziland, Botswana, and South Africa exceed 0·1% prevalence.[54]
country
vasectomy usage
notes
Canada 22% "of all women rely on vasectomy"
UK 17% - 21% only range given
New Zealand 17% to 21% only range given
South Korea 17% to 21% only range given
Australia ~10%
Belgium ~10%
Denmark ~10%
Spain ~10%
Switzerland ~10%
Swaziland 0.3%
Botswana 0.4%
South Africa 0.7%
In North America and Europe vasectomy usage is on the order of 10% with some countries reaching 20%.[54] Despite its high efficacy, in the United States, vasectomy is utilized less than half the rate of the alternative female tubal ligation.[56] According to the research, vasectomy is least utilized among black and Latin populations, the U.S. groups that have the highest rates of female sterilization.[56]
New Zealand, in contrast to the U.S., has higher levels of vasectomy than tubal ligation uptake. 18% of all men, and 25% of all married men have had a vasectomy. The age cohort with the highest level of vasectomy was 40–49, where 57% of men had taken it up.[57] Canada, the UK, Bhutan and the Netherlands all have similar levels of uptake.[58]
History[edit]
The first recorded vasectomy was performed on a dog in 1823.[59] A short time after that, R. Harrison of London performed the first human vasectomy; however the surgery was not done for sterilization purposes, but to bring about atrophy of the prostate.[citation needed] Soon, however, it was believed to have benefits for eugenics. The first case report of vasectomy in the U.S. was in 1897, by A. J. Ochsner, a surgeon in Chicago, in a paper titled, "Surgical treatment of habitual criminals." He believed vasectomy to be a simple, effective means for stemming the tide of racial degeneration widely believed to be occurring.[citation needed] In 1902, Harry C. Sharp, the surgeon at the Indiana Reformatory, reported that he had sterilized 42 inmates, in an effort to both reduce criminal behavior in those individuals and prevent the birth of future criminals.[60]
However, some vasectomies were not performed with the goal of sterilization in mind. Eugen Steinach (1861–1944), an Austrian physician, believed that a unilateral vasectomy (severing only one of the two vas deferentia) in older men could restore general vigor and sexual potency, shrink enlarged prostates, and cure various ailments by somehow boosting the hormonal output of the vasectomied testicle.[61] This surgery, which became very popular in the 1920s, was undertaken by many wealthy men, including Sigmund Freud and W. B. Yeats.[62] Since these operations lacked rigorous controlled trials, any rejuvenating effect was likely due to the Placebo Effect, and with the later development of synthetic injectable hormones, this operation fell out of vogue.[61][63]
Vasectomy began to be regarded as a method of consensual birth control during the Second World War.[citation needed] The first vasectomy program on a national scale was launched in 1954 in India.[64]
Society and culture[edit]
Availability and legality[edit]
Vasectomy costs are (or may be) covered in different countries, as a method of both contraception or population control, with some offering it as a part of a national health insurance. The Affordable Care Act of the U.S. does not cover vasectomy. Vasectomy was generally considered illegal in France until 2001, due to provisions in the Napoleonic Code forbidding "self-mutilation". No French law specifically mentioned vasectomy until a 2001 law on contraception and foeticide permitted the procedure.[65]
Ideological issues[edit]
The emphasis on "shared responsibility" has been taken up in recent research and articles by Terry and Braun, who regard much of the earlier psychological research on vasectomy as seemingly negative, or 'suspicious' in tone.[66] In research based on 16 New Zealand men (chosen for their enthusiasm on the topic of vasectomy), researchers extracted primary themes from their interviews of "taking responsibility" and "vasectomy as an act of minor heroism".[67]
The need to "target men’s involvement in reproductive and contraceptive practices" was historically raised on a global scale at the 1994 International Conference on Population and Development (ICPD) in Cairo,[68] in relation to both population control and decreasing the levels of inequality in the ‘contraceptive burden’, which has traditionally placed responsibility for contraception unfairly upon women. Vigoya has referred to a global "cultura anticonceptiva femenina"—a female contraceptive culture, where, despite the possibility of men taking more responsibility for contraception, there is virtually nowhere in the world where true contraceptive equality exists.[69]
Critical health researchers emphasize the positive identities that men can take up post vasectomy, as a "man who takes on responsibility for the contraceptive task"[67] and a man who is willing to "sacrifice" his fertility for his partner and family's sake.[66] Often these sorts of accounts are constructed within the 'contraceptive economy' of a relationship, in which women have maintained responsibility of the contraceptive task up until the point of the operation. Terry notes that a man undergoing a vasectomy may also mean he receives a high degree of gratitude and positive reinforcement for making the choice to be sterilised, perhaps more so than a woman who has been on an oral contraceptive or similar for years prior.[67]
Tourism[edit]
Main article: Medical tourism
Medical tourism, where a patient travels to a less developed location where a procedure is cheaper to save money and combine convalescence with a vacation, is infrequently used for vasectomy due to its low cost, but is more likely to be used for vasectomy reversal. Many overseas hospitals list vasectomy as one of their qualified surgical procedures. Medical tourism has come under the scrutiny of some governments for quality of care and postoperative care issues.[70]
See also[edit]
Male contraceptive
Sperm granuloma
Testicular sperm extraction
Orchiectomy
List of surgeries by type
World Vasectomy Day
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Jump up ^ Köhler, TS.; Choy, JT.; Fazili, AA.; Koenig, JF.; Brannigan, RE. (Nov 2012). "A critical analysis of the reported association between vasectomy and frontotemporal dementia". Asian J Androl. 14 (6): 903–4. doi:10.1038/aja.2012.94. PMC 3720109  . PMID 23064682.
^ Jump up to: a b Rogalski E, Weintraub S, Mesulam MM (2013). "Are there susceptibility factors for primary progressive aphasia?". Brain Lang. 127 (2): 135–8. doi:10.1016/j.bandl.2013.02.004. PMC 3740011  . PMID 23489582.
^ Jump up to: a b Cook, LA; Pun, A; Gallo, MF; Lopez, LM; Van Vliet, HA (30 March 2014). "Scalpel versus no-scalpel incision for vasectomy". The Cochrane Database of Systematic Reviews (3): CD004112. doi:10.1002/14651858.CD004112.pub4. PMID 24683021.
Jump up ^ webmaster@vasectomy-information.com (2007-09-14). "Recanalization of the vas deferens". Vasectomy-information.com. Retrieved 2011-12-28.
Jump up ^ Sokal, David; Irsula, Belinda; Hays, Melissa; Chen-Mok, Mario; Barone, Mark A; Investigator Study, Group (2004). "Vasectomy by ligation and excision, with or without fascial interposition: a randomized controlled trial ISRCTN77781689". BMC Medicine. 2 (1): 6. doi:10.1186/1741-7015-2-6. PMC 406425  . PMID 15056388.
^ Jump up to: a b Weiss, RS; Li, PS (2005). "No-needle jet anesthetic technique for no-scalpel vasectomy". The Journal of Urology. 173 (5): 1677–80. doi:10.1097/01.ju.0000154698.03817.d4. PMID 15821547.
Jump up ^ Panel Members: Ira D. Sharlip, M.D., Panel Co-Chair Clinical Professor Department of Urology University of California, San Francisco, CA San Francisco, CA Arnold M. Belker, M.D., Panel Co-Chair Emeritus Clinical Professor Department of Urology University of Louisville School of Medicine Louisville, Kentucky Stanton Honig, M.D. Professor of Surgery/Urology University of Connecticut Farmington CT The Urology Center New Haven, CT Michel Labrecque, M.D., Ph.D. Professor Department of Family and Emergency Medicine Université Laval Quebec City, Canada Joel L. Marmar, M.D. Professor of Urology Cooper Medical School of Rowan University Camden, NJ Lawrence S. Ross, M.D. Clarence C. Saelhof Professor Emeritus Department of Urology University of Illinois at Chicago Chicago, IL Jay I. Sandlow, M.D. Professor of Urology Medical College of Wisconsin Milwaukee, WI David C. Sokal, MD Senior Scientist Clinical Sciences Department FHI 360 Durham, NC (January 2015). "American Urology Association Vasectomy guideline 2012 (amended 2015)" (PDF). American Urology Association. American Urological Association Education and Research. Archived from the original (PDF) on 29 January 2017. Retrieved 1 February 2017. CS1 maint: Multiple names: authors list (link)
Jump up ^ Moss, WM (December 1992). "A comparison of open-end versus closed-end vasectomies: a report on 6220 cases". Contraception. 46 (6): 521–5. doi:10.1016/0010-7824(92)90116-B. PMID 1493712.
Jump up ^ Shapiro, EI; Silber, SJ (November 1979). "Open-ended vasectomy, sperm granuloma, and postvasectomy orchialgia". Fertility and Sterility. 32 (5): 546–50. PMID 499585.
^ Jump up to: a b Cook, Lynley A.; Van Vliet, Huib AAM; Lopez, Laureen M; Pun, Asha; Gallo, Maria F (2007). Cook, Lynley A., ed. "Vasectomy occlusion techniques for male sterilization". Cochrane Database of Systematic Reviews. doi:10.1002/14651858.CD003991.pub3.
Jump up ^ Levine, LA; Abern, MR; Lux, MM (2006). "Persistent motile sperm after ligation band vasectomy". The Journal of Urology. 176 (5): 2146–8. doi:10.1016/j.juro.2006.07.028. PMID 17070280.
Jump up ^ "Male Contraception Update for the public - August 2008, 3(8)". Imccoalition.org. Archived from the original on 2012-03-27. Retrieved 2011-12-28.
^ Jump up to: a b http://www.vasweb.com/vasclip.htm[unreliable medical source?]
Jump up ^ "Injected plugs". MaleContraceptives.org. 2011-07-27. Retrieved 2011-12-28.
Jump up ^ "Intra Vas Device (IVD)". MaleContraceptives.org. Retrieved 2011-12-28.
Jump up ^ Healthwise Staff (May 13, 2010). "Vasectomy Procedure, Effects, Risks, Effectiveness, and More". webmd.com. Retrieved March 29, 2012.
Jump up ^ "Post Vasectomy Semen Analysis". Cambridge IVF. Cambridge IVF. Retrieved 23 October 2015.
^ Jump up to: a b Murphy, Clare. "Divorce fuels vasectomy reversals", BBC News, 18 March 2009. Retrieved 19 September 2012.
Jump up ^ Shridharani, Anand; Sandlow, Jay I (2010). "Vasectomy reversal versus IVF with sperm retrieval: which is better?". Current Opinion in Urology. 20 (6): 503–509. doi:10.1097/MOU.0b013e32833f1b35. PMID 20852426.
Jump up ^ Landry E, Ward V (1997). "Perspectives from Couples on the Vasectomy Decision: A Six-Country Study" (PDF). Reproductive Health Matters. (special issue): 58–67.
Jump up ^ Jamieson, D (2002). "A comparison of women's regret after vasectomy versus tubal sterilization". Obstetrics & Gynecology. 99 (6): 1073–1079. doi:10.1016/S0029-7844(02)01981-6.
Jump up ^ "About Vasectomy Reversal". Professor Earl Owen's homepage. Archived from the original on 2007-12-06. Retrieved 2007-11-29.
Jump up ^ Owen ER (1977). "Microsurgical vasovasostomy: a reliable vasectomy reversal". Urology. 167 (2 Pt 2): 1205. doi:10.1016/S0022-5347(02)80388-3. PMID 11905902.
Jump up ^ Vasectomy Reversal http://www.epigee.org/guide/vasectomy_reversal.html
Jump up ^ Cited in Laurance, Jeremy (2009) "Vasectomy Reversal: First Cut Isn't Final" in The Independent 30 March 2009 https://www.independent.co.uk/life-style/health-and-families/health-news/vasectomy-reversal-first-cut-isnt-final-1657039.html
^ Jump up to: a b Sukcharoen, Nares; Ngeamvijawat, J; Sithipravej, T; Promviengchai, S (2003). "High sex chromosome aneuploidy and diploidy rate of epididymal spermatozoa in obstructive azoospermic men". Journal of Assisted Reproduction and Genetics. 20 (5): 196–203. doi:10.1023/A:1023674110940. PMC 3455301  . PMID 12812463.
Jump up ^ Abdelmassih, V.; Balmaceda, JP; Tesarik, J; Abdelmassih, R; Nagy, ZP (2002). "Relationship between time period after vasectomy and the reproductive capacity of sperm obtained by epididymal aspiration". Human Reproduction. 17 (3): 736–740. doi:10.1093/humrep/17.3.736. PMID 11870128.
Jump up ^ Horovitz, D. (February 2012). "Vasectomy reversal provides long-term pain relief for men with the post-vasectomy pain syndrome". National Center for Biotechnology Information, U.S. National Library of Medicine. 187: 613–7. doi:10.1016/j.juro.2011.10.023. PMID 22177173.
^ Jump up to: a b c Jacobstein, Roy (December 2015). "The kindest cut: global need to increase vasectomy availability". The Lancet. Global Health. 3 (12): e733–734. doi:10.1016/S2214-109X(15)00168-0. ISSN 2214-109X. PMID 26545447.
Jump up ^ "Contraceptive Use 2011". UN Department of Economic and Social Affairs, Population Division, 2012.
^ Jump up to: a b Shih G, Turok DK, Parker WJ (April 2011). "Vasectomy: the other (better) form of sterilization". Contraception. 83 (4): 310–5. doi:10.1016/j.contraception.2010.08.019. PMID 21397087.
Jump up ^ Sneyd, Mary Jane; Cox, Brian; Paul, Charlotte; Skegg, David C.G. (2001). "High prevalence of vasectomy in New Zealand". Contraception. 64 (3): 155–159. doi:10.1016/S0010-7824(01)00242-6.
Jump up ^ Pile, John M.; Barone, Mark A. (2009). "Demographics of Vasectomy—USA and International". Urologic Clinics of North America. 36 (3): 295–305. doi:10.1016/j.ucl.2009.05.006. PMID 19643232.
Jump up ^ Leavesley, JH (1980). "Brief history of vasectomy". Family planning information service. 1 (5): 2–3. PMID 12336890.
Jump up ^ Reilly, Phillip (1991). The Surgical Solution: A History of Involuntary Sterilization in the United States. Johns Hopkins University Press. pp. 30–33.
^ Jump up to: a b Schultheiss, Dirk; Engel, Rainer M. (2003-11-01). "G. Frank Lydston (1858–1923) revisited: androgen therapy by testicular implantation in the early twentieth century". World Journal of Urology. 21 (5): 356–363. doi:10.1007/s00345-003-0370-z. ISSN 0724-4983.
Jump up ^ Ellmann, Richard. "Yeats's Second Puberty". The New York Review of Books. Retrieved 2017-08-23.
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Jump up ^ Sharma, Sanjay (April–June 2014). "A Study of Male Sterilization with No Scalpel Vasectomy" (PDF). JK Science. 16 (2): 67. Retrieved 23 October 2015.
Jump up ^ Latham, Melanie (2002-07-05). Regulating reproduction: a century of conflict in Britain and France. Manchester University Press. ISBN 0-7190-5699-3.
^ Jump up to: a b Terry, Gareth; Braun, Virginia (2011). "'I'm committed to her and the family': positive accounts of vasectomy among New Zealand men". Journal of Reproductive and Infant Psychology. 29 (3): 276–291. doi:10.1080/02646838.2011.592976.
^ Jump up to: a b c Terry, G.; Braun, V. (2011). "'It's kind of me taking responsibility for these things': Men, vasectomy and 'contraceptive economies'" (PDF). Feminism & Psychology. 21 (4): 477–495. doi:10.1177/0959353511419814.
Jump up ^ Mundigo A (2000) Re-conceptualizing the role of men in the post-Cairo era" Culture, Health & Sexuality 2(3) 323–337; cited in Terry, G and Braun, V (2011). 'It’s kind of me taking responsibility for these things': men, vasectomy and 'contraceptive economies' Feminism & Psychology 21(4), pp. 477–495. Available online at:http://oro.open.ac.uk/34285/1/It%27s%20kind%20of%20me%20taking%20responsbility%20for%20these%20things.pdf
Jump up ^ cited in Terry, G and Braun, V (2011). 'It’s kind of me taking responsibility for these things': men, vasectomy and 'contraceptive economies' Feminism & Psychology 21(4), p.1 Available online at: http://oro.open.ac.uk/34285/1/It%27s%20kind%20of%20me%20taking%20responsbility%20for%20these%20things.pdf
Jump up ^ Lunt, Neil; Carrera, Percivil (2010). "Medical tourism: Assessing the evidence on treatment abroad". Maturitas. 66 (1): 27–32. doi:10.1016/j.maturitas.2010.01.017. PMID 20185254.
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Natural number
From Wikipedia, the free encyclopedia
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This article is about “positive integers” and “non-negative integers”. For all the numbers …, −2, −1, 0, 1, 2, …, see Integer.
Natural numbers can be used for counting (one apple, two apples, three apples, …)
Real numbers (ℝ) include the rational (ℚ), which include the integers (ℤ), which include the natural numbers (ℕ)
In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country"). In common language, words used for counting are "cardinal numbers" and words used for ordering are "ordinal numbers".
Some definitions, including the standard ISO 80000-2,[1] begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, ….[2][3][4][5] Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers (including negative integers).[6]
The natural numbers are the basis from which many other number sets may be built by extension: the integers, by including (if not yet in) the neutral element 0 and an additive inverse (−n) for each nonzero natural number n; the rational numbers, by including a multiplicative inverse (1/n) for each nonzero integer n (and also the product of these inverses by integers); the real numbers by including with the rationals the limits of (converging) Cauchy sequences of rationals; the complex numbers, by including with the real numbers the unresolved square root of minus one (and also the sums and products thereof); and so on.[7][8] These chains of extensions make the natural numbers canonically embedded (identified) in the other number systems.
Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics.
In common language, for example in primary school, natural numbers may be called counting numbers[9] both to intuitively exclude the negative integers and zero, and also to contrast the discreteness of counting to the continuity of measurement, established by the real numbers.
The natural numbers can, at times, appear as a convenient set of names (labels), that is, as what linguists call nominal numbers, foregoing many or all of the properties of being a number in a mathematical sense.
Contents
[hide]
1 History
1.1 Ancient roots
1.2 Modern definitions
2 Notation
3 Properties
3.1 Addition
3.2 Multiplication
3.3 Relationship between addition and multiplication
3.4 Order
3.5 Division
3.6 Algebraic properties satisfied by the natural numbers
4 Generalizations
5 Formal definitions
5.1 Peano axioms
5.2 Constructions based on set theory
5.2.1 Von Neumann construction
5.2.2 Other constructions
6 See also
7 Notes
8 References
9 External links
History[edit]
Ancient roots[edit]
The Ishango bone (on exhibition at the Royal Belgian Institute of Natural Sciences)[10][11][12] is believed to have been used 20,000 years ago for natural number arithmetic.
The most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set.
The first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers. The ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, and 6 ones; and similarly for the number 4,622. The Babylonians had a place-value system based essentially on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context.[13]
A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation (within other numbers) dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number.[14] The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica.[15][16] The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus (the calculation of the date of Easter), beginning with Dionysius Exiguus in 525, without being denoted by a numeral (standard Roman numerals do not have a symbol for 0); instead nulla (or the genitive form nullae) from nullus, the Latin word for "none", was employed to denote a 0 value.[17]
The first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes even not as a number at all.[18]
Independent studies also occurred at around the same time in India, China, and Mesoamerica.[19]
Modern definitions[edit]
This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (October 2014) (Learn how and when to remove this template message)
In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school[which?] of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man".[20]
In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics.[21] In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural but a consequence of definitions. Later, two classes of such formal definitions were constructed; later, they were shown to be equivalent in most practical applications.
Set-theoretical definitions of natural numbers were initiated by Frege and he initially defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox. Therefore, this formalism was modified so that a natural number is defined as a particular set, and any set that can be put into one-to-one correspondence with that set is said to have that number of elements.[22]
The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic. It is based on an axiomatization of the properties of ordinal numbers: each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several weak systems of set theory. One such system is ZFC with the axiom of infinity replaced by its negation. Theorems that can be proved in ZFC but cannot be proved using the Peano Axioms include Goodstein's theorem.[23]
With all these definitions it is convenient to include 0 (corresponding to the empty set) as a natural number. Including 0 is now the common convention among set theorists[24] and logicians.[25] Other mathematicians also include 0[5] although many have kept the older tradition and take 1 to be the first natural number.[26] Computer scientists often start from zero when enumerating items like loop counters and string- or array- elements.[27][28]
Notation[edit]
The double-struck capital N symbol, often used to denote the set of all natural numbers (see List of mathematical symbols).
Mathematicians use N or ℕ (an N in blackboard bold) to refer to the set of all natural numbers. Older texts have also occasionally employed J as the symbol for this set.[29] This set is countably infinite: it is infinite but countable by definition. This is also expressed by saying that the cardinal number of the set is aleph-naught (ℵ0).[30]
To be unambiguous about whether 0 is included or not, sometimes an index (or superscript) "0" is added in the former case, and a superscript "*" or subscript ">0" is added in the latter case:[1]
ℕ0 = ℕ0 = {0, 1, 2, …}
ℕ* = ℕ+ = ℕ1 = ℕ>0 = {1, 2, …}.
Alternatively, natural numbers may be distinguished from positive integers with the index notation, but it must be understood by context that since both symbols are used, the natural numbers contain zero.[31]
ℕ = {0, 1, 2, …}.
ℤ+= {1, 2, …}.
Properties[edit]
Addition[edit]
One can recursively define an addition operator on the natural numbers by setting a + 0 = a and a + S(b) = S(a + b) for all a, b. Here S should be read as "successor". This turns the natural numbers (ℕ, +) into a commutative monoid with identity element 0, the so-called free object with one generator. This monoid satisfies the cancellation property and can be embedded in a group (in the mathematical sense of the word group). The smallest group containing the natural numbers is the integers.
If 1 is defined as S(0), then b + 1 = b + S(0) = S(b + 0) = S(b). That is, b + 1 is simply the successor of b.
Multiplication[edit]
Analogously, given that addition has been defined, a multiplication operator × can be defined via a × 0 = 0 and a × S(b) = (a × b) + a. This turns (ℕ*, ×) into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers.
Relationship between addition and multiplication[edit]
Addition and multiplication are compatible, which is expressed in the distribution law: a × (b + c) = (a × b) + (a × c). These properties of addition and multiplication make the natural numbers an instance of a commutative semiring. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. The lack of additive inverses, which is equivalent to the fact that ℕ is not closed under subtraction (i.e., subtracting one natural from another does not always result in another natural), means that ℕ is not a ring; instead it is a semiring (also known as a rig).
If the natural numbers are taken as "excluding 0", and "starting at 1", the definitions of + and × are as above, except that they begin with a + 1 = S(a) and a × 1 = a.
Order[edit]
In this section, juxtaposed variables such as ab indicate the product a × b, and the standard order of operations is assumed.
A total order on the natural numbers is defined by letting a ≤ b if and only if there exists another natural number c where a + c = b. This order is compatible with the arithmetical operations in the following sense: if a, b and c are natural numbers and a ≤ b, then a + c ≤ b + c and ac ≤ bc.
An important property of the natural numbers is that they are well-ordered: every non-empty set of natural numbers has a least element. The rank among well-ordered sets is expressed by an ordinal number; for the natural numbers, this is denoted as ω (omega).
Division[edit]
In this section, juxtaposed variables such as ab indicate the product a × b, and the standard order of operations is assumed.
While it is in general not possible to divide one natural number by another and get a natural number as result, the procedure of division with remainder is available as a substitute: for any two natural numbers a and b with b ≠ 0 there are natural numbers q and r such that
a = bq + r and r < b.
The number q is called the quotient and r is called the remainder of the division of a by b. The numbers q and r are uniquely determined by a and b. This Euclidean division is key to several other properties (divisibility), algorithms (such as the Euclidean algorithm), and ideas in number theory.
Algebraic properties satisfied by the natural numbers[edit]
The addition (+) and multiplication (×) operations on natural numbers as defined above have several algebraic properties:
Closure under addition and multiplication: for all natural numbers a and b, both a + b and a × b are natural numbers.
Associativity: for all natural numbers a, b, and c, a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c.
Commutativity: for all natural numbers a and b, a + b = b + a and a × b = b × a.
Existence of identity elements: for every natural number a, a + 0 = a and a × 1 = a.
Distributivity of multiplication over addition for all natural numbers a, b, and c, a × (b + c) = (a × b) + (a × c).
No nonzero zero divisors: if a and b are natural numbers such that a × b = 0, then a = 0 or b = 0 (or both).
Generalizations[edit]
Two important generalizations of natural numbers arise from the two uses of counting and ordering: cardinal numbers and ordinal numbers.
A natural number can be used to express the size of a finite set; more precisely, a cardinal number is a measure for the size of a set, which is even suitable for infinite sets. This concept of "size" relies on maps between sets, such that two sets have the same size, exactly if there exists a bijection between them. The set of natural numbers itself, and any bijective image of it, is said to be countably infinite and to have cardinality aleph-null (ℵ0).
Natural numbers are also used as linguistic ordinal numbers: "first", "second", "third", and so forth. This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of any well-ordered countably infinite set. This assignment can be generalized to general well-orderings with a cardinality beyond countability, to yield the ordinal numbers. An ordinal number may also be used to describe the notion of "size" for a well-ordered set, in a sense different from cardinality: if there is an order isomorphism (more than a bijection!) between two well-ordered sets, they have the same ordinal number. The first ordinal number that is not a natural number is expressed as ω; this is also the ordinal number of the set of natural numbers itself.
Many well-ordered sets with cardinal number ℵ0 have an ordinal number greater than ω (the latter is the lowest possible). The least ordinal of cardinality ℵ0 (i.e., the initial ordinal) is ω.
For finite well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural number, the number of elements of the set. This number can also be used to describe the position of an element in a larger finite, or an infinite, sequence.
A countable non-standard model of arithmetic satisfying the Peano Arithmetic (i.e., the first-order Peano axioms) was developed by Skolem in 1933. The hypernatural numbers are an uncountable model that can be constructed from the ordinary natural numbers via the ultrapower construction.
Georges Reeb used to claim provocatively that The naïve integers don't fill up ℕ. Other generalizations are discussed in the article on numbers.
Formal definitions[edit]
Peano axioms[edit]
Main article: Peano axioms
Many properties of the natural numbers can be derived from the Peano axioms.[32][33]
Axiom One: 0 is a natural number.
Axiom Two: Every natural number has a successor.
Axiom Three: 0 is not the successor of any natural number.
Axiom Four: If the successor of x {\displaystyle x} equals the successor of y {\displaystyle y} , then x {\displaystyle x} equals y {\displaystyle y} .
Axiom Five (the axiom of induction): If a statement is true of 0, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number.
These are not the original axioms published by Peano, but are named in his honor. Some forms of the Peano axioms have 1 in place of 0. In ordinary arithmetic, the successor of x {\displaystyle x} is x + 1 {\displaystyle x+1} . Replacing Axiom Five by an axiom schema one obtains a (weaker) first-order theory called Peano Arithmetic.
Constructions based on set theory[edit]
Main article: Set-theoretic definition of natural numbers
Von Neumann construction[edit]
In the area of mathematics called set theory, a special case of the von Neumann ordinal construction [34] defines the natural numbers as follows:
Set 0 = { }, the empty set,
Define S(a) = a ∪ {a} for every set a. S(a) is the successor of a, and S is called the successor function.
By the axiom of infinity, there exists a set which contains 0 and is closed under the successor function. Such sets are said to be 'inductive'. The intersection of all such inductive sets is defined to be the set of natural numbers. It can be checked that the set of natural numbers satisfies the Peano axioms.
It follows that each natural number is equal to the set of all natural numbers less than it:
0 = { },
1 = 0 ∪ {0} = {0} = {{ }},
2 = 1 ∪ {1} = {0, 1} = {{ }, {{ }}},
3 = 2 ∪ {2} = {0, 1, 2} = {{ }, {{ }}, {{ }, {{ }}}},
n = n−1 ∪ {n−1} = {0, 1, …, n−1} = {{ }, {{ }}, …, {{ }, {{ }}, …}}, etc.
With this definition, a natural number n is a particular set with n elements, and n ≤ m if and only if n is a subset of m.
Also, with this definition, different possible interpretations of notations like ℝn (n-tuples versus mappings of n into ℝ) coincide.
Even if one does not accept the axiom of infinity and therefore cannot accept that the set of all natural numbers exists, it is still possible to define any one of these sets.
Other constructions[edit]
Although the standard construction is useful, it is not the only possible construction. Zermelo's construction goes as follows:
Set 0 = { }
Define S(a) = {a},
It then follows that
0 = { },
1 = {0} = {{ }},
2 = {1} = {{{ }}},
n = {n−1} = {{{…}}}, etc.
Each natural number is then equal to the set containing just the natural number preceding it.
See also[edit]
Mathematics portal
Integer
Set-theoretic definition of natural numbers
Peano axioms
Canonical representation of a positive integer
Countable set
Number#Classification for other number systems (rational, real, complex etc.)
Notes[edit]
^ Jump up to: a b "Standard number sets and intervals". ISO 80000-2:2009. International Organization for Standardization. p. 6.
Jump up ^ Weisstein, Eric W. "Natural Number". MathWorld.
Jump up ^ "natural number", Merriam-Webster.com, Merriam-Webster, retrieved 4 October 2014
Jump up ^ Carothers (2000) says: "ℕ is the set of natural numbers (positive integers)" (p. 3)
^ Jump up to: a b Mac Lane & Birkhoff (1999) include zero in the natural numbers: 'Intuitively, the set ℕ = {0, 1, 2, ...} of all natural numbers may be described as follows: ℕ contains an "initial" number 0; ...'. They follow that with their version of the Peano Postulates. (p. 15)
Jump up ^ Jack G. Ganssle & Michael Barr (2003). Embedded Systems Dictionary. p. 138 (integer), 247 (signed integer), & 276 (unsigned integer). ISBN 1578201209. integer 1. n. Any whole number.
Jump up ^ Mendelson (2008) says: "The whole fantastic hierarchy of number systems is built up by purely set-theoretic means from a few simple assumptions about natural numbers." (Preface, p. x)
Jump up ^ Bluman (2010): "Numbers make up the foundation of mathematics." (p. 1)
Jump up ^ Weisstein, Eric W. "Counting Number". MathWorld.
Jump up ^ Introduction, Royal Belgian Institute of Natural Sciences, Brussels, Belgium.
Jump up ^ Flash presentation, Royal Belgian Institute of Natural Sciences, Brussels, Belgium.
Jump up ^ The Ishango Bone, Democratic Republic of the Congo, on permanent display at the Royal Belgian Institute of Natural Sciences, Brussels, Belgium. UNESCO's Portal to the Heritage of Astronomy
Jump up ^ Georges Ifrah, The Universal History of Numbers, Wiley, 2000, ISBN 0-471-37568-3
Jump up ^ "A history of Zero". MacTutor History of Mathematics. Retrieved 2013-01-23. … a tablet found at Kish … thought to date from around 700 BC, uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place
Jump up ^ Mann, Charles C. (2005), 1491: New Revelations Of The Americas Before Columbus, Knopf, p. 19, ISBN 9781400040063 .
Jump up ^ Evans, Brian (2014), "Chapter 10. Pre-Columbian Mathematics: The Olmec, Maya, and Inca Civilizations", The Development of Mathematics Throughout the Centuries: A Brief History in a Cultural Context, John Wiley & Sons, ISBN 9781118853979 .
Jump up ^ Michael L. Gorodetsky (2003-08-25). "Cyclus Decemnovennalis Dionysii – Nineteen year cycle of Dionysius". Hbar.phys.msu.ru. Retrieved 2012-02-13.
Jump up ^ This convention is used, for example, in Euclid's Elements, see Book VII, definitions 1 and 2.
Jump up ^ Morris Kline, Mathematical Thought From Ancient to Modern Times, Oxford University Press, 1990 [1972], ISBN 0-19-506135-7
Jump up ^ The English translation is from Gray. In a footnote, Gray attributes the German quote to: "Weber 1891/92, 19, quoting from a lecture of Kronecker's of 1886."
Gray, Jeremy (2008), Plato's Ghost: The Modernist Transformation of Mathematics, Princeton University Press, p. 153
Weber, Heinrich L. 1891-2. Kronecker. Jahresbericht der Deutschen Mathematiker-Vereinigung 2:5-23. (The quote is on p. 19.)
Jump up ^ "Much of the mathematical work of the twentieth century has been devoted to examining the logical foundations and structure of the subject." (Eves 1990, p. 606)
Jump up ^ Eves 1990, Chapter 15
Jump up ^ L. Kirby; J. Paris, Accessible Independence Results for Peano Arithmetic, Bulletin of the London Mathematical Society 14 (4): 285. doi:10.1112/blms/14.4.285, 1982.
Jump up ^ Bagaria, Joan. "Set Theory". The Stanford Encyclopedia of Philosophy (Winter 2014 Edition).
Jump up ^ Goldrei, Derek (1998). "3". Classic set theory : a guided independent study (1. ed., 1. print ed.). Boca Raton, Fla. [u.a.]: Chapman & Hall/CRC. p. 33. ISBN 0-412-60610-0.
Jump up ^ This is common in texts about Real analysis. See, for example, Carothers (2000, p. 3) or Thomson, Bruckner & Bruckner (2000, p. 2).
Jump up ^ Brown, Jim (1978). "In Defense of Index Origin 0". ACM SIGAPL APL Quote Quad. 9 (2): 7. doi:10.1145/586050.586053. Retrieved 19 January 2015.
Jump up ^ Hui, Roger. "Is Index Origin 0 a Hindrance?". jsoftware.com. Retrieved 19 January 2015.
Jump up ^ Rudin, W. (1976). Principles of Mathematical Analysis (PDF). New York: McGraw-Hill. p. 25. ISBN 978-0-07-054235-8.
Jump up ^ Weisstein, Eric W. "Cardinal Number". MathWorld.
Jump up ^ Grimaldi, Ralph P. (2003). A review of discrete and combinatorial mathematics (5th ed.). Boston, MA: Addison-Wesley. p. 133. ISBN 978-0201726343.
Jump up ^ G.E. Mints (originator), "Peano axioms", Encyclopedia of Mathematics, Springer, in cooperation with the European Mathematical Society, retrieved 8 October 2014
Jump up ^ Hamilton (1988) calls them "Peano's Postulates" and begins with "1. 0 is a natural number." (p. 117f)
Halmos (1960) uses the language of set theory instead of the language of arithmetic for his five axioms. He begins with "(I) 0 ∈ ω (where, of course, 0 = ∅" (ω is the set of all natural numbers). (p. 46)
Morash (1991) gives "a two-part axiom" in which the natural numbers begin with 1. (Section 10.1: An Axiomatization for the System of Positive Integers)
Jump up ^ Von Neumann 1923
References[edit]
Bluman, Allan (2010), Pre-Algebra DeMYSTiFieD (Second ed.), McGraw-Hill Professional
Carothers, N.L. (2000), Real analysis, Cambridge University Press, ISBN 0-521-49756-6
Clapham, Christopher; Nicholson, James (2014), The Concise Oxford Dictionary of Mathematics (Fifth ed.), Oxford University Press
Dedekind, Richard (1963), Essays on the Theory of Numbers, Dover, ISBN 0-486-21010-3
Dedekind, Richard (2007), Essays on the Theory of Numbers, Kessinger Publishing, LLC, ISBN 0-548-08985-X
Eves, Howard (1990), An Introduction to the History of Mathematics (6th ed.), Thomson, ISBN 978-0-03-029558-4
Halmos, Paul (1960), Naive Set Theory, Springer Science & Business Media
Hamilton, A. G. (1988), Logic for Mathematicians (Revised ed.), Cambridge University Press
James, Robert C.; James, Glenn (1992), Mathematics Dictionary (Fifth ed.), Chapman & Hall
Landau, Edmund (1966), Foundations of Analysis (Third ed.), Chelsea Pub Co, ISBN 0-8218-2693-X
Mac Lane, Saunders; Birkhoff, Garrett (1999), Algebra (3rd ed.), American Mathematical Society
Mendelson, Elliott (2008) [1973], Number Systems and the Foundations of Analysis, Dover Publications
Morash, Ronald P. (1991), Bridge to Abstract Mathematics: Mathematical Proof and Structures (Second ed.), Mcgraw-Hill College
Musser, Gary L.; Peterson, Blake E.; Burger, William F. (2013), Mathematics for Elementary Teachers: A Contemporary Approach (10th ed.), Wiley Global Education, ISBN 978-1118457443
Szczepanski, Amy F.; Kositsky, Andrew P. (2008), The Complete Idiot's Guide to Pre-algebra, Penguin Group
Thomson, Brian S.; Bruckner, Judith B.; Bruckner, Andrew M. (2008), Elementary Real Analysis (Second ed.), ClassicalRealAnalysis.com, ISBN 9781434843678
Von Neumann, Johann (1923), "Zur Einführung der transfiniten Zahlen", Acta litterarum ac scientiarum Ragiae Universitatis Hungaricae Francisco-Josephinae, Sectio scientiarum mathematicarum, 1: 199–208
Von Neumann, John (January 2002) [1923], "On the introduction of transfinite numbers", in Jean van Heijenoort, From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931 (3rd ed.), Harvard University Press, pp. 346–354, ISBN 0-674-32449-8 - English translation of von Neumann 1923.
External links[edit]
Hazewinkel, Michiel, ed. (2001) [1994], "Natural number", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
Axioms and Construction of Natural Numbers
Essays on the Theory of Numbers by Richard Dedekind at Project Gutenberg
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Recursively defined
numbers
Fibonacci
Jacobsthal
Leonardo
Lucas
Padovan
Pell
Perrin
Possessing a
specific set
of other numbers
Knödel
Riesel
Sierpinski
Expressible via
specific sums
Nonhypotenuse
Polite
Practical
Primary pseudoperfect
Ulam
Wolstenholme
Generated
via a sieve
Lucky
Code related
Meertens
Figurate numbers
2-dimensional
centered
Centered triangular
Centered square
Centered pentagonal
Centered hexagonal
Centered heptagonal
Centered octagonal
Centered nonagonal
Centered decagonal
Star
non-centered
Triangular
Square
Square triangular
Pentagonal
Hexagonal
Heptagonal
Octagonal
Nonagonal
Decagonal
Dodecagonal
3-dimensional
centered
Centered tetrahedral
Centered cube
Centered octahedral
Centered dodecahedral
Centered icosahedral
non-centered
Tetrahedral
Octahedral
Dodecahedral
Icosahedral
Stella octangula
pyramidal
Square pyramidal
Pentagonal pyramidal
Hexagonal pyramidal
Heptagonal pyramidal
4-dimensional
centered
Centered pentachoric
Squared triangular
non-centered
Pentatope
Pseudoprimes
Carmichael number
Catalan pseudoprime
Elliptic pseudoprime
Euler pseudoprime
Euler–Jacobi pseudoprime
Fermat pseudoprime
Frobenius pseudoprime
Lucas pseudoprime
Somer–Lucas pseudoprime
Strong pseudoprime
Combinatorial
numbers
Bell
Cake
Catalan
Dedekind
Delannoy
Euler
Fuss–Catalan
Lazy caterer's sequence
Lobb
Motzkin
Narayana
Ordered Bell
Schröder
Schröder–Hipparchus
Arithmetic functions
By properties of σ(n)
Abundant
Almost perfect
Arithmetic
Colossally abundant
Descartes
Hemiperfect
Highly abundant
Highly composite
Hyperperfect
Multiply perfect
Perfect
Practical number
Primitive abundant
Quasiperfect
Refactorable
Sublime
Superabundant
Superior highly composite
Superperfect
By properties of Ω(n)
Almost prime
Semiprime
By properties of φ(n)
Highly cototient
Highly totient
Noncototient
Nontotient
Perfect totient
Sparsely totient
By properties of s(n)
Amicable
Betrothed
Deficient
Semiperfect
Euclid
Fortunate
Dividing a quotient
Wieferich
Wall–Sun–Sun
Wolstenholme prime
Wilson
Other prime factor
or divisor related
numbers
Blum
Erdős–Nicolas
Erdős–Woods
Friendly
Frugal
Giuga
Harmonic divisor
Lucas–Carmichael
Pronic
Regular
Rough
Smooth
Sociable
Sphenic
Størmer
Super-Poulet
Zeisel
Recreational
mathematics
Base-dependent
numbers
Automorphic
Cyclic
Digit-reassembly
Dudeney
Equidigital
Extravagant
Factorion
Friedman
Happy
Harshad
Kaprekar
Keith
Lychrel
Missing-digit sum
Narcissistic
Palindromic
Pandigital
Parasitic
Pernicious
Polydivisible
Primeval
Repdigit
Repunit
Self
Self-descriptive
Smarandache–Wellin
Strictly non-palindromic
Strobogrammatic
Sum-product
Transposable
Trimorphic
Undulating
Vampire
Aronson's sequence
Ban
Pancake number
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