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Raw Queries and Responses

1. Shard Calculation

Query:

Response:

Calculated Shard: 70 (from laksa022)

2. Crawled Status Check

Query:

curl -X POST \
  'http://laksa070.int.ahrefs:8124/' \
  -H 'Content-Type: text/plain' \
  -H 'X-ClickHouse-Database: crawler3' \
  -H 'Authorization: Basic YXBpOg==' \
  -d 'SELECT getAhrefsURLFromUnparsed(src_unparsed) AS found_url, ifNull(toUnixTimestamp(download_stamp), 0) AS crawl_time, ifNull(toUnixTimestamp(props_url_first_seen), 0) AS first_indexed_time, download_http_code AS http_code, src_unparsed AS src_unparsed, src_root_hash AS src_root_hash, history_drop_reason AS history_drop_reason, meta_title AS meta_title, meta_descriptions AS meta_descriptions, attrs_boilerpipe_text AS attrs_boilerpipe_text, attrs_markdown AS attrs_markdown, attrs_readable_markdown AS attrs_readable_markdown, meta_canonical AS meta_canonical FROM crawler3.page_info_local FINAL PREWHERE (src_root_hash, src_unparsed) IN ((getAhrefsRootHashFromUnparsed(getAhrefsUnparsedNoserviceFromURL(\'https://www.sciencenews.org/article/eulers-beautiful-equation\')), getAhrefsUnparsedNoserviceFromURL(\'https://www.sciencenews.org/article/eulers-beautiful-equation\'))) FORMAT JSONEachRow'

Response:

{"found_url":"https:\/\/www.sciencenews.org\/article\/eulers-beautiful-equation","crawl_time":1775939550,"first_indexed_time":1542768996,"http_code":200,"src_unparsed":"org,sciencenews!www,\/article\/eulers-beautiful-equation s443","src_root_hash":"11628747150769427070","history_drop_reason":null,"meta_title":"Euler's Beautiful Equation","meta_descriptions":["Leonhard Euler, one of the greatest mathematicians of all time, was born 300 years ago on April 15, 1707."],"attrs_boilerpipe_text":"The famous mathematician Leonhard Euler was born 300 years ago.\nPortrait by Johann Georg Brucker\/Wikipedia\nGraphic by Spencer Norcross for Science News\n“Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means. But we have proved it, and therefore we know it must be the truth\n.” —Benjamin Pierce, a Harvard mathematician, after proving Euler’s equation,\ne\ni\np\n= –1, in a 19th-century lecture.\nSunday, April 15, is the 300th birthday of Leonhard Euler (pronounced “oiler”), one of the most important mathematicians ever to have lived. His works help form the foundation of nearly all areas of mathematics, including calculus, number theory, geometry, and applied math.\nOne of the many discoveries for which he is famous is the equation\ne\nip\n= –1 . In a 1988 poll, readers of the journal\nMathematical Intelligencer\nchose this equation as the single most beautiful equation in all of mathematics. The equation weaves together four seemingly unrelated mathematical numbers,\ne\n, p,\ni\n, and –1, in an astonishingly simple way.\nBut what does\ne\nip\n= –1 really mean?\nFirst, let’s examine what the letters mean. The symbol\ne\nstands for a particular irrational number. Since it is irrational, its value can’t be given precisely in decimal notation, but it is approximately equal to 2.7183. Euler introduced this constant to the world of mathematics. He probably named it after the word “exponential,” because\ne\nis the base of the natural logarithms. Initially, he recognized the importance of\ne\nbecause of its remarkable properties in calculus. But\ne\npops up over and over again in surprising places throughout mathematics. Somehow, this nearly magical number seems to tie the world together.\nPi, or p, is another irrational number. It rounds off to 3.14159, and it is defined as the ratio of the circumference of a circle to its diameter.\nThe “imaginary” number\ni\nis defined as the square root of –1. Imaginary numbers are unlike any number we encounter in ordinary experience. If\ni\nwere an ordinary positive number, then multiplying it by itself would give a positive number, not –1. And if\ni\nwere an ordinary negative number, then multiplying it by itself would also give a positive number, because multiplying a negative number by another negative number produces a positive number. Mathematicians therefore invented imaginary numbers, and they gave the name\ni\nto the square root of –1.\nCooking up new numbers might seem like a questionable proposition. But when a strange creation like\ni\nturns out to have remarkable and surprising properties, such as linking\ne\nto p in a simple equation, mathematicians are inclined to put aside any lingering qualms in favor of investigating what other secrets\ni\nmight be hiding.\nImagining the imaginary\nOn inspection,\ne\nip\nis a peculiar concept. If the existence of an imaginary number seems strange, raising a real number to the\npower\nof an imaginary number is even odder. What could it possibly mean?\nEuler found an answer to that question almost accidentally. He set out to understand functions, which are essentially mathematical machines that start with one number and produce another. Mathematical functions include\nx\n2\n+ 3\nx\n+ 1, and sin\nx\n, and\ne\nx\n. Each of those formulas provides a method for starting with one number to produce another number.\nEuler built on a remarkable discovery that mathematician Brook Taylor made in 1712. Taylor discovered that all types of mathematical functions, no matter how diverse they may seem, can be expressed in the same form, as a sum of powers of\nx\n.\nWhen we apply Taylor’s method to the function\ne\nx\n, we get the following equation:\nThe right-hand side of the equation looks much more complicated than the left, but in fact it is much easier to calculate. The sums of powers of\nx\n, which we call polynomials, are the simplest kind of function in mathematics. In this case, the equation is a bit more complex than an ordinary polynomial because the sum continues infinitely. Nevertheless, expressing a function in this form allows mathematicians to understand it more clearly. Most importantly, the equation’s right-hand side doesn’t have any number raised to the power of\nx\n.\nTaylor’s expansion offers a way to understand the complex and peculiar expression\ne\nip\nfrom Euler’s equation. That expression raises\ne\nto the power of\ni\n, which is a perplexing concept. But using the expanded form—the right side of Taylor’s equation—helps clarify matters.\nEuler also faced the unfortunate fact that the Taylor expansion of\ne\nip\ncontinues infinitely. He solved that problem by comparing the Taylor expansion of\ne\nix\nwith those of cos\nx\nand sin\nx\n. Those functions seem to be completely unrelated to\ne\nx\n, since they describe the relationships between the lengths of the sides of triangles. But Euler found the following remarkable identity:\ne\nix\n= cos\nx\n+\ni\nsin\nx\nWhen Euler set\nx\nequal to p, this equation made it easy to calculate the value of\ne\nip\n. The cosine of p equals –1, and the sine of p equals 0. So\ne\nin\n= (–1) +\ni\n(0). Therefore,\ne\nip\n= –1 + 0 = –1.\nIn honor of Euler’s birthday, we can all enjoy a birthday gift from him: Euler’s beautiful theorem.\nEditor’s note: The character that appears as “p” on this page represents the Greek letter pi\n. In some browsers, it may not display properly.\nIf you would like to comment on this article, please see the\nblog version\n.","attrs_markdown":"[Skip to content](https:\/\/www.sciencenews.org\/article\/eulers-beautiful-equation#content)\n\n**Subscribe today**\n\nEvery print subscription comes with full digital access\n\n[Subscribe Now](https:\/\/www.sciencenews.org\/subscribe1?key=9MNAVBNA&utm_source=banner&utm_medium=site&utm_campaign=full_digital_access)\n\nMenu\n\n- [All Topics](https:\/\/www.sciencenews.org\/topics\/)\n  - [Health](https:\/\/www.sciencenews.org\/topic\/health-medicine)\n  - [Humans](https:\/\/www.sciencenews.org\/topic\/humans)\n    - [Anthropology](https:\/\/www.sciencenews.org\/topic\/anthropology)\n    - [Health & Medicine](https:\/\/www.sciencenews.org\/topic\/health-medicine)\n    - [Archaeology](https:\/\/www.sciencenews.org\/topic\/archaeology)\n    - [Psychology](https:\/\/www.sciencenews.org\/topic\/psychology)\n    - [View All](https:\/\/www.sciencenews.org\/topic\/humans)\n  - 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More than **2 years ago**\n\n## Share this:\n- [Share](https:\/\/www.sciencenews.org\/article\/eulers-beautiful-equation)\n\n- [Share via email (Opens in new window) Email](mailto:?subject=Euler%27s%20Beautiful%20Equation&body=I%20saw%20this%20on%20Science%20News%3A%20https%3A%2F%2Fwww.sciencenews.org%2Farticle%2Feulers-beautiful-equation%3Futm_source%3Dinternal%26utm_medium%3Demail%26utm_campaign%3Demail_share)\n- [Share on Facebook (Opens in new window) Facebook](https:\/\/www.sciencenews.org\/article\/eulers-beautiful-equation?share=facebook)\n- [Share on Reddit (Opens in new window) Reddit](https:\/\/www.sciencenews.org\/article\/eulers-beautiful-equation?share=reddit)\n- [Share on X (Opens in new window) X](https:\/\/www.sciencenews.org\/article\/eulers-beautiful-equation?share=twitter)\n- [Print (Opens in new window) Print](https:\/\/www.sciencenews.org\/article\/eulers-beautiful-equation#print?share=print)\n\n![](https:\/\/www.sciencenews.org\/wp-content\/uploads\/sites\/2\/2007\/04\/6684.jpg)\n\nThe famous mathematician Leonhard Euler was born 300 years ago. Portrait by Johann Georg Brucker\/Wikipedia\n\n![](https:\/\/www.sciencenews.org\/wp-content\/uploads\/sites\/2\/2007\/04\/6685.jpg)\n\nGraphic by Spencer Norcross for Science News\n\n> *“Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means. But we have proved it, and therefore we know it must be the truth*.” —Benjamin Pierce, a Harvard mathematician, after proving Euler’s equation, *ei[p](https:\/\/www.sciencenews.org\/article\/eulers-beautiful-equation#pi)* = –1, in a 19th-century lecture.\n\nSunday, April 15, is the 300th birthday of Leonhard Euler (pronounced “oiler”), one of the most important mathematicians ever to have lived. His works help form the foundation of nearly all areas of mathematics, including calculus, number theory, geometry, and applied math.\n\n### Sign up for our newsletter\nWe summarize the week's scientific breakthroughs every Thursday.\n\nOne of the many discoveries for which he is famous is the equation *eip* = –1 . In a 1988 poll, readers of the journal *Mathematical Intelligencer* chose this equation as the single most beautiful equation in all of mathematics. The equation weaves together four seemingly unrelated mathematical numbers, *e*, p, *i*, and –1, in an astonishingly simple way.\n\nBut what does *eip* = –1 really mean?\n\nFirst, let’s examine what the letters mean. The symbol *e* stands for a particular irrational number. Since it is irrational, its value can’t be given precisely in decimal notation, but it is approximately equal to 2.7183. Euler introduced this constant to the world of mathematics. He probably named it after the word “exponential,” because *e* is the base of the natural logarithms. Initially, he recognized the importance of *e* because of its remarkable properties in calculus. But *e* pops up over and over again in surprising places throughout mathematics. Somehow, this nearly magical number seems to tie the world together.\n\nPi, or p, is another irrational number. It rounds off to 3.14159, and it is defined as the ratio of the circumference of a circle to its diameter.\n\nThe “imaginary” number *i* is defined as the square root of –1. Imaginary numbers are unlike any number we encounter in ordinary experience. If *i* were an ordinary positive number, then multiplying it by itself would give a positive number, not –1. And if *i* were an ordinary negative number, then multiplying it by itself would also give a positive number, because multiplying a negative number by another negative number produces a positive number. Mathematicians therefore invented imaginary numbers, and they gave the name *i* to the square root of –1.\n\nSponsor Message\n\nCooking up new numbers might seem like a questionable proposition. But when a strange creation like *i* turns out to have remarkable and surprising properties, such as linking *e* to p in a simple equation, mathematicians are inclined to put aside any lingering qualms in favor of investigating what other secrets *i* might be hiding.\n\n**Imagining the imaginary**\n\nOn inspection, *eip* is a peculiar concept. If the existence of an imaginary number seems strange, raising a real number to the *power* of an imaginary number is even odder. What could it possibly mean?\n\nEuler found an answer to that question almost accidentally. He set out to understand functions, which are essentially mathematical machines that start with one number and produce another. Mathematical functions include *x*2 + 3*x* + 1, and sin *x*, and *ex*. Each of those formulas provides a method for starting with one number to produce another number.\n\nEuler built on a remarkable discovery that mathematician Brook Taylor made in 1712. Taylor discovered that all types of mathematical functions, no matter how diverse they may seem, can be expressed in the same form, as a sum of powers of *x*.\n\nWhen we apply Taylor’s method to the function *ex*, we get the following equation:\n\nThe right-hand side of the equation looks much more complicated than the left, but in fact it is much easier to calculate. The sums of powers of *x*, which we call polynomials, are the simplest kind of function in mathematics. In this case, the equation is a bit more complex than an ordinary polynomial because the sum continues infinitely. Nevertheless, expressing a function in this form allows mathematicians to understand it more clearly. Most importantly, the equation’s right-hand side doesn’t have any number raised to the power of *x*.\n\nTaylor’s expansion offers a way to understand the complex and peculiar expression *eip* from Euler’s equation. That expression raises *e* to the power of *i*, which is a perplexing concept. But using the expanded form—the right side of Taylor’s equation—helps clarify matters.\n\nEuler also faced the unfortunate fact that the Taylor expansion of *eip* continues infinitely. He solved that problem by comparing the Taylor expansion of *eix* with those of cos *x* and sin *x*. Those functions seem to be completely unrelated to *ex*, since they describe the relationships between the lengths of the sides of triangles. But Euler found the following remarkable identity:\n\n*eix* = cos *x* + *i* sin *x*\n\nWhen Euler set *x* equal to p, this equation made it easy to calculate the value of *eip*. The cosine of p equals –1, and the sine of p equals 0. So *ein* = (–1) + *i* (0). Therefore, *eip* = –1 + 0 = –1.\n\nIn honor of Euler’s birthday, we can all enjoy a birthday gift from him: Euler’s beautiful theorem.\n\nEditor’s note: The character that appears as “p” on this page represents the Greek letter pi*. In some browsers, it may not display properly.*\n***\nIf you would like to comment on this article, please see the [blog version](http:\/\/blog.sciencenews.org\/mathtrek\/2007\/04\/eulers_beautiful_equation.html).\n\nQuestions or comments on this article? E-mail us at [feedback@sciencenews.org](mailto:feedback@sciencenews.org) \\| [Reprints FAQ](https:\/\/www.sciencenews.org\/permission-republish)\n\nWe are at a critical time and ***supporting climate journalism is more important than ever.** Science News and our parent organization, the Society for Science, need your help to strengthen environmental literacy and ensure that our response to climate change is informed by science.*\n\nPlease  [subscribe to *Science News* and **add \\$16**](https:\/\/www.sciencenews.org\/subscribe1-print-digital?key=9MENDTEST&utm_source=end-of-article&utm_medium=site&utm_campaign=non-sub-control) to expand science literacy and understanding.\n\n### Popular Stories\n1. 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All rights reserved.\n\n1776 Massachusetts Ave NW, Washington, DC 20036 [202\\.785.2255](tel:202.785.2255)\n\n[Terms of Service](https:\/\/www.sciencenews.org\/terms-of-service) [Privacy Policy](https:\/\/www.sciencenews.org\/privacy-policy)\n\nPrivacy Manager\n\nClose\n\n## Log in\nSubscribers, enter your e-mail address for full access to the *Science News* archives and digital editions.\n\nNot a subscriber?  \n [Become one now](https:\/\/www.sciencenews.org\/subscription?utm_source=sciencenews&utm_medium=login&utm_campaign=sn-login&tfa_3134=sn-login).","attrs_readable_markdown":"![](https:\/\/www.sciencenews.org\/wp-content\/uploads\/sites\/2\/2007\/04\/6684.jpg)\n\nThe famous mathematician Leonhard Euler was born 300 years ago. Portrait by Johann Georg Brucker\/Wikipedia\n\n![](https:\/\/www.sciencenews.org\/wp-content\/uploads\/sites\/2\/2007\/04\/6685.jpg)\n\nGraphic by Spencer Norcross for Science News\n\n> *“Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means. But we have proved it, and therefore we know it must be the truth*.” —Benjamin Pierce, a Harvard mathematician, after proving Euler’s equation, *ei[p](https:\/\/www.sciencenews.org\/article\/eulers-beautiful-equation#pi)* = –1, in a 19th-century lecture.\n\nSunday, April 15, is the 300th birthday of Leonhard Euler (pronounced “oiler”), one of the most important mathematicians ever to have lived. His works help form the foundation of nearly all areas of mathematics, including calculus, number theory, geometry, and applied math.\n\nOne of the many discoveries for which he is famous is the equation *eip* = –1 . In a 1988 poll, readers of the journal *Mathematical Intelligencer* chose this equation as the single most beautiful equation in all of mathematics. The equation weaves together four seemingly unrelated mathematical numbers, *e*, p, *i*, and –1, in an astonishingly simple way.\n\nBut what does *eip* = –1 really mean?\n\nFirst, let’s examine what the letters mean. The symbol *e* stands for a particular irrational number. Since it is irrational, its value can’t be given precisely in decimal notation, but it is approximately equal to 2.7183. Euler introduced this constant to the world of mathematics. He probably named it after the word “exponential,” because *e* is the base of the natural logarithms. Initially, he recognized the importance of *e* because of its remarkable properties in calculus. But *e* pops up over and over again in surprising places throughout mathematics. Somehow, this nearly magical number seems to tie the world together.\n\nPi, or p, is another irrational number. It rounds off to 3.14159, and it is defined as the ratio of the circumference of a circle to its diameter.\n\nThe “imaginary” number *i* is defined as the square root of –1. Imaginary numbers are unlike any number we encounter in ordinary experience. If *i* were an ordinary positive number, then multiplying it by itself would give a positive number, not –1. And if *i* were an ordinary negative number, then multiplying it by itself would also give a positive number, because multiplying a negative number by another negative number produces a positive number. Mathematicians therefore invented imaginary numbers, and they gave the name *i* to the square root of –1.\n\nCooking up new numbers might seem like a questionable proposition. But when a strange creation like *i* turns out to have remarkable and surprising properties, such as linking *e* to p in a simple equation, mathematicians are inclined to put aside any lingering qualms in favor of investigating what other secrets *i* might be hiding.\n\n**Imagining the imaginary**\n\nOn inspection, *eip* is a peculiar concept. If the existence of an imaginary number seems strange, raising a real number to the *power* of an imaginary number is even odder. What could it possibly mean?\n\nEuler found an answer to that question almost accidentally. He set out to understand functions, which are essentially mathematical machines that start with one number and produce another. Mathematical functions include *x*2 + 3*x* + 1, and sin *x*, and *ex*. Each of those formulas provides a method for starting with one number to produce another number.\n\nEuler built on a remarkable discovery that mathematician Brook Taylor made in 1712. Taylor discovered that all types of mathematical functions, no matter how diverse they may seem, can be expressed in the same form, as a sum of powers of *x*.\n\nWhen we apply Taylor’s method to the function *ex*, we get the following equation:\n\nThe right-hand side of the equation looks much more complicated than the left, but in fact it is much easier to calculate. The sums of powers of *x*, which we call polynomials, are the simplest kind of function in mathematics. In this case, the equation is a bit more complex than an ordinary polynomial because the sum continues infinitely. Nevertheless, expressing a function in this form allows mathematicians to understand it more clearly. Most importantly, the equation’s right-hand side doesn’t have any number raised to the power of *x*.\n\nTaylor’s expansion offers a way to understand the complex and peculiar expression *eip* from Euler’s equation. That expression raises *e* to the power of *i*, which is a perplexing concept. But using the expanded form—the right side of Taylor’s equation—helps clarify matters.\n\nEuler also faced the unfortunate fact that the Taylor expansion of *eip* continues infinitely. He solved that problem by comparing the Taylor expansion of *eix* with those of cos *x* and sin *x*. Those functions seem to be completely unrelated to *ex*, since they describe the relationships between the lengths of the sides of triangles. But Euler found the following remarkable identity:\n\n*eix* = cos *x* + *i* sin *x*\n\nWhen Euler set *x* equal to p, this equation made it easy to calculate the value of *eip*. The cosine of p equals –1, and the sine of p equals 0. So *ein* = (–1) + *i* (0). Therefore, *eip* = –1 + 0 = –1.\n\nIn honor of Euler’s birthday, we can all enjoy a birthday gift from him: Euler’s beautiful theorem.\n\nEditor’s note: The character that appears as “p” on this page represents the Greek letter pi*. In some browsers, it may not display properly.*\n***\nIf you would like to comment on this article, please see the [blog version](http:\/\/blog.sciencenews.org\/mathtrek\/2007\/04\/eulers_beautiful_equation.html).","meta_canonical":null}

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Boilerpipe Text	The famous mathematician Leonhard Euler was born 300 years ago. Portrait by Johann Georg Brucker/Wikipedia Graphic by Spencer Norcross for Science News “Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means. But we have proved it, and therefore we know it must be the truth .” —Benjamin Pierce, a Harvard mathematician, after proving Euler’s equation, e i p = –1, in a 19th-century lecture. Sunday, April 15, is the 300th birthday of Leonhard Euler (pronounced “oiler”), one of the most important mathematicians ever to have lived. His works help form the foundation of nearly all areas of mathematics, including calculus, number theory, geometry, and applied math. One of the many discoveries for which he is famous is the equation e ip = –1 . In a 1988 poll, readers of the journal Mathematical Intelligencer chose this equation as the single most beautiful equation in all of mathematics. The equation weaves together four seemingly unrelated mathematical numbers, e , p, i , and –1, in an astonishingly simple way. But what does e ip = –1 really mean? First, let’s examine what the letters mean. The symbol e stands for a particular irrational number. Since it is irrational, its value can’t be given precisely in decimal notation, but it is approximately equal to 2.7183. Euler introduced this constant to the world of mathematics. He probably named it after the word “exponential,” because e is the base of the natural logarithms. Initially, he recognized the importance of e because of its remarkable properties in calculus. But e pops up over and over again in surprising places throughout mathematics. Somehow, this nearly magical number seems to tie the world together. Pi, or p, is another irrational number. It rounds off to 3.14159, and it is defined as the ratio of the circumference of a circle to its diameter. The “imaginary” number i is defined as the square root of –1. Imaginary numbers are unlike any number we encounter in ordinary experience. If i were an ordinary positive number, then multiplying it by itself would give a positive number, not –1. And if i were an ordinary negative number, then multiplying it by itself would also give a positive number, because multiplying a negative number by another negative number produces a positive number. Mathematicians therefore invented imaginary numbers, and they gave the name i to the square root of –1. Cooking up new numbers might seem like a questionable proposition. But when a strange creation like i turns out to have remarkable and surprising properties, such as linking e to p in a simple equation, mathematicians are inclined to put aside any lingering qualms in favor of investigating what other secrets i might be hiding. Imagining the imaginary On inspection, e ip is a peculiar concept. If the existence of an imaginary number seems strange, raising a real number to the power of an imaginary number is even odder. What could it possibly mean? Euler found an answer to that question almost accidentally. He set out to understand functions, which are essentially mathematical machines that start with one number and produce another. Mathematical functions include x 2 + 3 x + 1, and sin x , and e x . Each of those formulas provides a method for starting with one number to produce another number. Euler built on a remarkable discovery that mathematician Brook Taylor made in 1712. Taylor discovered that all types of mathematical functions, no matter how diverse they may seem, can be expressed in the same form, as a sum of powers of x . When we apply Taylor’s method to the function e x , we get the following equation: The right-hand side of the equation looks much more complicated than the left, but in fact it is much easier to calculate. The sums of powers of x , which we call polynomials, are the simplest kind of function in mathematics. In this case, the equation is a bit more complex than an ordinary polynomial because the sum continues infinitely. Nevertheless, expressing a function in this form allows mathematicians to understand it more clearly. Most importantly, the equation’s right-hand side doesn’t have any number raised to the power of x . Taylor’s expansion offers a way to understand the complex and peculiar expression e ip from Euler’s equation. That expression raises e to the power of i , which is a perplexing concept. But using the expanded form—the right side of Taylor’s equation—helps clarify matters. Euler also faced the unfortunate fact that the Taylor expansion of e ip continues infinitely. He solved that problem by comparing the Taylor expansion of e ix with those of cos x and sin x . Those functions seem to be completely unrelated to e x , since they describe the relationships between the lengths of the sides of triangles. But Euler found the following remarkable identity: e ix = cos x + i sin x When Euler set x equal to p, this equation made it easy to calculate the value of e ip . The cosine of p equals –1, and the sine of p equals 0. So e in = (–1) + i (0). Therefore, e ip = –1 + 0 = –1. In honor of Euler’s birthday, we can all enjoy a birthday gift from him: Euler’s beautiful theorem. Editor’s note: The character that appears as “p” on this page represents the Greek letter pi . In some browsers, it may not display properly. If you would like to comment on this article, please see the blog version .
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Portrait by Johann Georg Brucker/Wikipedia ![](https://www.sciencenews.org/wp-content/uploads/sites/2/2007/04/6685.jpg) Graphic by Spencer Norcross for Science News > “Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means. But we have proved it, and therefore we know it must be the truth.” —Benjamin Pierce, a Harvard mathematician, after proving Euler’s equation, ei[p](https://www.sciencenews.org/article/eulers-beautiful-equation#pi) = –1, in a 19th-century lecture. Sunday, April 15, is the 300th birthday of Leonhard Euler (pronounced “oiler”), one of the most important mathematicians ever to have lived. His works help form the foundation of nearly all areas of mathematics, including calculus, number theory, geometry, and applied math. ### Sign up for our newsletter We summarize the week's scientific breakthroughs every Thursday. One of the many discoveries for which he is famous is the equation eip = –1 . In a 1988 poll, readers of the journal Mathematical Intelligencer chose this equation as the single most beautiful equation in all of mathematics. The equation weaves together four seemingly unrelated mathematical numbers, e, p, i, and –1, in an astonishingly simple way. But what does eip = –1 really mean? First, let’s examine what the letters mean. The symbol e stands for a particular irrational number. Since it is irrational, its value can’t be given precisely in decimal notation, but it is approximately equal to 2.7183. Euler introduced this constant to the world of mathematics. He probably named it after the word “exponential,” because e is the base of the natural logarithms. Initially, he recognized the importance of e because of its remarkable properties in calculus. But e pops up over and over again in surprising places throughout mathematics. Somehow, this nearly magical number seems to tie the world together. Pi, or p, is another irrational number. It rounds off to 3.14159, and it is defined as the ratio of the circumference of a circle to its diameter. The “imaginary” number i is defined as the square root of –1. Imaginary numbers are unlike any number we encounter in ordinary experience. If i were an ordinary positive number, then multiplying it by itself would give a positive number, not –1. And if i were an ordinary negative number, then multiplying it by itself would also give a positive number, because multiplying a negative number by another negative number produces a positive number. Mathematicians therefore invented imaginary numbers, and they gave the name i to the square root of –1. Sponsor Message Cooking up new numbers might seem like a questionable proposition. But when a strange creation like i turns out to have remarkable and surprising properties, such as linking e to p in a simple equation, mathematicians are inclined to put aside any lingering qualms in favor of investigating what other secrets i might be hiding. Imagining the imaginary On inspection, eip is a peculiar concept. If the existence of an imaginary number seems strange, raising a real number to the power of an imaginary number is even odder. What could it possibly mean? Euler found an answer to that question almost accidentally. He set out to understand functions, which are essentially mathematical machines that start with one number and produce another. Mathematical functions include x2 + 3x + 1, and sin x, and ex. Each of those formulas provides a method for starting with one number to produce another number. Euler built on a remarkable discovery that mathematician Brook Taylor made in 1712. Taylor discovered that all types of mathematical functions, no matter how diverse they may seem, can be expressed in the same form, as a sum of powers of x. When we apply Taylor’s method to the function ex, we get the following equation: The right-hand side of the equation looks much more complicated than the left, but in fact it is much easier to calculate. The sums of powers of x, which we call polynomials, are the simplest kind of function in mathematics. In this case, the equation is a bit more complex than an ordinary polynomial because the sum continues infinitely. Nevertheless, expressing a function in this form allows mathematicians to understand it more clearly. Most importantly, the equation’s right-hand side doesn’t have any number raised to the power of x. Taylor’s expansion offers a way to understand the complex and peculiar expression eip from Euler’s equation. That expression raises e to the power of i, which is a perplexing concept. But using the expanded form—the right side of Taylor’s equation—helps clarify matters. Euler also faced the unfortunate fact that the Taylor expansion of eip continues infinitely. He solved that problem by comparing the Taylor expansion of eix with those of cos x and sin x. Those functions seem to be completely unrelated to ex, since they describe the relationships between the lengths of the sides of triangles. But Euler found the following remarkable identity: eix = cos x + i sin x When Euler set x equal to p, this equation made it easy to calculate the value of eip. The cosine of p equals –1, and the sine of p equals 0. So ein = (–1) + i (0). Therefore, eip = –1 + 0 = –1. In honor of Euler’s birthday, we can all enjoy a birthday gift from him: Euler’s beautiful theorem. Editor’s note: The character that appears as “p” on this page represents the Greek letter pi. 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Readable Markdown	![](https://www.sciencenews.org/wp-content/uploads/sites/2/2007/04/6684.jpg) The famous mathematician Leonhard Euler was born 300 years ago. Portrait by Johann Georg Brucker/Wikipedia ![](https://www.sciencenews.org/wp-content/uploads/sites/2/2007/04/6685.jpg) Graphic by Spencer Norcross for Science News > “Gentlemen, that is surely true, it is absolutely paradoxical; we cannot understand it, and we don’t know what it means. But we have proved it, and therefore we know it must be the truth.” —Benjamin Pierce, a Harvard mathematician, after proving Euler’s equation, ei[p](https://www.sciencenews.org/article/eulers-beautiful-equation#pi) = –1, in a 19th-century lecture. Sunday, April 15, is the 300th birthday of Leonhard Euler (pronounced “oiler”), one of the most important mathematicians ever to have lived. His works help form the foundation of nearly all areas of mathematics, including calculus, number theory, geometry, and applied math. One of the many discoveries for which he is famous is the equation eip = –1 . In a 1988 poll, readers of the journal Mathematical Intelligencer chose this equation as the single most beautiful equation in all of mathematics. The equation weaves together four seemingly unrelated mathematical numbers, e, p, i, and –1, in an astonishingly simple way. But what does eip = –1 really mean? First, let’s examine what the letters mean. The symbol e stands for a particular irrational number. Since it is irrational, its value can’t be given precisely in decimal notation, but it is approximately equal to 2.7183. Euler introduced this constant to the world of mathematics. He probably named it after the word “exponential,” because e is the base of the natural logarithms. Initially, he recognized the importance of e because of its remarkable properties in calculus. But e pops up over and over again in surprising places throughout mathematics. Somehow, this nearly magical number seems to tie the world together. Pi, or p, is another irrational number. It rounds off to 3.14159, and it is defined as the ratio of the circumference of a circle to its diameter. The “imaginary” number i is defined as the square root of –1. Imaginary numbers are unlike any number we encounter in ordinary experience. If i were an ordinary positive number, then multiplying it by itself would give a positive number, not –1. And if i were an ordinary negative number, then multiplying it by itself would also give a positive number, because multiplying a negative number by another negative number produces a positive number. Mathematicians therefore invented imaginary numbers, and they gave the name i to the square root of –1. Cooking up new numbers might seem like a questionable proposition. But when a strange creation like i turns out to have remarkable and surprising properties, such as linking e to p in a simple equation, mathematicians are inclined to put aside any lingering qualms in favor of investigating what other secrets i might be hiding. Imagining the imaginary On inspection, eip is a peculiar concept. If the existence of an imaginary number seems strange, raising a real number to the power of an imaginary number is even odder. What could it possibly mean? Euler found an answer to that question almost accidentally. He set out to understand functions, which are essentially mathematical machines that start with one number and produce another. Mathematical functions include x2 + 3x + 1, and sin x, and ex. Each of those formulas provides a method for starting with one number to produce another number. Euler built on a remarkable discovery that mathematician Brook Taylor made in 1712. Taylor discovered that all types of mathematical functions, no matter how diverse they may seem, can be expressed in the same form, as a sum of powers of x. When we apply Taylor’s method to the function ex, we get the following equation: The right-hand side of the equation looks much more complicated than the left, but in fact it is much easier to calculate. The sums of powers of x, which we call polynomials, are the simplest kind of function in mathematics. In this case, the equation is a bit more complex than an ordinary polynomial because the sum continues infinitely. Nevertheless, expressing a function in this form allows mathematicians to understand it more clearly. Most importantly, the equation’s right-hand side doesn’t have any number raised to the power of x. Taylor’s expansion offers a way to understand the complex and peculiar expression eip from Euler’s equation. That expression raises e to the power of i, which is a perplexing concept. But using the expanded form—the right side of Taylor’s equation—helps clarify matters. Euler also faced the unfortunate fact that the Taylor expansion of eip continues infinitely. He solved that problem by comparing the Taylor expansion of eix with those of cos x and sin x. Those functions seem to be completely unrelated to ex, since they describe the relationships between the lengths of the sides of triangles. But Euler found the following remarkable identity: eix = cos x + i sin x When Euler set x equal to p, this equation made it easy to calculate the value of eip. The cosine of p equals –1, and the sine of p equals 0. So ein = (–1) + i (0). Therefore, eip = –1 + 0 = –1. In honor of Euler’s birthday, we can all enjoy a birthday gift from him: Euler’s beautiful theorem. Editor’s note: The character that appears as “p” on this page represents the Greek letter pi. In some browsers, it may not display properly. *** If you would like to comment on this article, please see the [blog version](http://blog.sciencenews.org/mathtrek/2007/04/eulers_beautiful_equation.html).
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