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...maller than zero). For example, 4 has two square roots: 2 and −2. The only square root of zero is zero. ...ry [[complex number]] except 0 has 2 square roots. For example: −1 has two square roots. We call them <math>i</math> and <math>-i</math>. ...

...l number instead of a positive real number. This is not possible with real numbers, as there is no real number that will multiply by itself to get a negative ...''' are to '''negative numbers''' what negative numbers are to '''positive numbers'''. If I say "go east by -1 mile" it is the same as if I had said "go west ...

...but in the [[integers]] subtraction does have closure. Subtraction of two numbers can produce a negative number, which is not a natural number, but it is an ...]] are a closure of the real numbers by including square roots of negative numbers. ...

There are a numbers of ways to calculate [[square root]]s of [[number]]s, and even more ways to [[Estimation|estimate]] them. ...ften [[square root]]s are [[Irrational numbers|irrational]]. For example, square root of a nice round [[whole number]] '''28''' is a [[Fraction (mathematics ...

...itself. 1, 4, 9, 16 and 25 are the first five [[square (algebra)|square]] numbers. ...>}} ([[exponentiation]]), usually pronounced as "''n'' squared". The name square number comes from the name of the shape; see [[#Properties|below]]. ...

...say "number" they usually mean "real number". The official symbol for real numbers is a bold '''R''' or a blackboard bold <math>\mathbb{R}</math>. ...ven minus signs (–) so that they are labeled differently from the positive numbers. ...

...[[infinite]] but if someone counted forever they wouldn't miss any of the numbers. Sometimes when people say 'countable set' they mean countable and infinit ...without missing any. The natural numbers have been nicknamed "the counting numbers" since they are what we usually use to count things with. ...

In [[math]], '''imaginary units''', or '''<math>i</math>''', are numbers that can be represented by equations but refer to values that could not phy ...e value of x^2 would have to equal -1. Though the problem is solvable, the square root of -1 could not be represented by a physical quantity of any objects i ...

...e a 41, and the mean does not tell us much about what kind of scores these numbers represent. To find the average of <math>N</math> numbers, the <math>N</math> numbers are [[addition|added]] and the total is [[Division (mathematics)|divide]]d ...

...ith special properties. The definition of the term is best understood for square matrices that are [[symmetry|symmetrical]], also known as [[Hermitian]] mat ...se]] and be greater than zero. The vector chosen must be filled with real numbers. ...

...ratio is ''a''/''b''. The person can make another ratio by adding the two numbers together ''a''+''b'' and dividing this by the larger number ''a''. The new The golden ratio is an [[Number#Irrational numbers|irrational number]]. If a person tries to write it, it will never stop and ...

Where the letters are the corresponding numbers of the original equation, {{math|1=''ax''² + ''bx'' + ''c'' = 0} The quadratic formula is proved by ''completing the square'', ...

...ue square added to the area of the red square makes the area of the purple square. It was named after the [[Ancient Greece|Greek]] [[mathematician]] [[Pythag ...ide]] of a [[square (geometry)|square]] has the same area as a half of the square. ...

...power is called '''square''' because the [[area]] of a [[square (geometry)|square]] is calculated using <math>a^2</math>. So :<math>x^2</math> is the square of <math>x</math> ...

...(number)|5]], and [[7 (number)|7]]. The next following eleven odd abundant numbers are *Odd abundant numbers below 500000 are in On-Line Encyclopedia of Integer Sequences [http://oeis. ...

...[linear algebra]], the '''[[identity matrix]]''' of size n is the n-by-n [[square matrix]] with ones on the main diagonal and zeros elsewhere. ...for all [[real number|real]] values of <math>\theta</math> (since the real numbers <math>\Bbb{R}</math> are the domain of sin and cos), as opposed to ...

...[induction]]. This is usually used to prove a theorem that is true for all numbers. There are 4 steps in a proof by induction. Prove that for all [[natural numbers]] ''n'', 2(1+2+3+....+''n''-1+''n'')=''n''(''n''+1) ...

|<center>'''Natural numbers example''' Numbers less than [[0]] (such as [[−1]]) are not natural numbers. ...

...of a matrix are often referenced by using pairs of [[subscript]]s, for the numbers at each of the '''rows''' & '''columns'''.]] ...th>B \cdot A </math>, which is the case for the multiplication of ordinary numbers. A matrix can have more than 2 [[dimension]]s, such as a 3D matrix. Also, a ...

...ve]] and [[distributive property|distributive properties]], just like real numbers. Complex numbers were invented to answer special equations that have [[exponent]]s in them. ...

:''For the book in the Bible, see [[Numbers (Bible)]]''. ...] or [[measurement|measure]]. Depending on the field of mathematics, where numbers are used, there are different definitions: ...

| [[square root]] | square root ...

It is a [[computation]] process that uses random numbers to produce an outcome(s). Instead of having fixed inputs, [[probability dis ...thematics)|π]]. A Monte Carlo algorithm would randomly place points in the square and use the percentage of points falling inside of the circle to estimate t ...

...is the name given by [[mathematics|mathematicians]] to a series of special numbers: 2, 1729 etc. A taxicab number is the smallest number that can be expresse ...y itself the answer is called a “square”, e.g. 3x3=9, so the number 9 is a square. ...

...</math>.<ref>Wells, D. ''The Penguin Dictionary of Curious and Interesting Numbers'' London: Penguin Group. (1987): 41–44</ref> An [[integer]] is ''[[even]]'' ...e]]. Two and three are the only prime numbers next to each other. The even numbers above two are not prime because they are divisible by 2. ...

.... The difference is called <math>d</math>, and the number you get when you square <math>d</math> is called <math>d^2</math>.<ref name=CGP-book/> <math>\sum d^2</math> means that we take the total of all the numbers that were in the column <math>d^2</math>. This is because <math>\sum</math> ...

...h>, then the time taken to produce the output would take a time that would square in magnitude with change in size of the input - for e.g. if someone gave an ...not talking about how the time changes when we're given a set of 10 or 20 numbers (in fact the big O, due to its mathematical definition, may even give wrong ...

...re very close to the [[average]]. A high standard deviation means that the numbers are spread out.<ref>{{cite journal ...context. [[science|Scientists]] commonly report the standard deviation of numbers from the average number in experiments. They often decide that only differe ...

...With sample standard deviation|sample standard deviation]]) divided by the square root of the sample size. ...t time period when the whole group covers a longer time. In this case, the numbers in the sample are not independent. Then special equations are used to try t ...

...point out the dot that is sometimes used to separate the positions of the numbers in this [[system]]. Almost everyone uses this nowadays and prefers the conv ...han ten distinct values (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) to represent any numbers, no matter how large. These digits are often used with a [[decimal separato ...

However, the value of K_c will depend on the ionic strength. (The square brackets mean the concentration of A, B and so on.) ...ame place to react in the opposite direction. The ratio between those two numbers represents how popular each side of the reaction will be when equilibrium i ...

...he number itself: 6) are 1, 2, and 3 and 1 + 2 + 3 equals 6. Other perfect numbers include 28, 496 and 8128. ==Perfect numbers that are even== ...

...ites, called [http://en.wikipedia.org/wiki/Carmichael_number '''Carmichael numbers'''], that may fool the Fermat test. The Carmichael numbers are very rare. Let <math>c(n)</math> be the "Carmichael density" that ...

...to solve real problems because the rules of algebra work in real life and numbers can be used to represent the values of real things. [[Physics]], [[enginee ...ns involve [[exponent]]s, starting with [[square (algebra)|squares]] and [[square root]]s. Interestingly, many of these rules can also be used on the variabl ...

...utation <math>\pi</math> of <math>n</math> numbers, we want to reorder the numbers in <math>\pi</math> in increasing order. This is the problem of sorting, on ...l numbers in <math>\pi'</math> are smaller than <math>\pi_i</math> and all numbers in <math>\pi''</math> are larger than <math>\pi_i</math>; ...

Adding the numbers on the left, subtracting <math>216.75</math> from both sides, and dividing Taking the square root of both sides gives us an answer of <math>v = \pm1.5</math>. ...

...</math> remains as a '''formal variable''' without assuming any value. The numbers that we want to count are the coefficients carried by the terms in the form === Fibonacci numbers === ...

...</math> remains as a '''formal variable''' without assuming any value. The numbers that we want to count are the coefficients carried by the terms in the form === Fibonacci numbers === ...

...</math> remains as a '''formal variable''' without assuming any value. The numbers that we want to count are the coefficients carried by the terms in the form === Fibonacci numbers === ...

...</math> remains as a '''formal variable''' without assuming any value. The numbers that we want to count are the coefficients carried by the terms in the form === Fibonacci numbers === ...

...</math> remains as a '''formal variable''' without assuming any value. The numbers that we want to count are the coefficients carried by the terms in the form === Fibonacci numbers === ...

...</math> remains as a '''formal variable''' without assuming any value. The numbers that we want to count are the coefficients carried by the terms in the form === Fibonacci numbers === ...

...</math> remains as a '''formal variable''' without assuming any value. The numbers that we want to count are the coefficients carried by the terms in the form === Fibonacci numbers === ...

...</math> remains as a '''formal variable''' without assuming any value. The numbers that we want to count are the coefficients carried by the terms in the form === Fibonacci numbers === ...

...</math> remains as a '''formal variable''' without assuming any value. The numbers that we want to count are the coefficients carried by the terms in the form === Fibonacci numbers === ...

...</math> remains as a '''formal variable''' without assuming any value. The numbers that we want to count are the coefficients carried by the terms in the form === Fibonacci numbers === ...

...n|exponential functions]], and are useful in multiplying or dividing large numbers. * If you have x²=3, then you use the (square) [[Root (mathematics)|root]] to find out x: You get the result x = <math di ...

...q i\leq n</math>, be independent, uniformly distributed points in the unit square <math>[0,1]^2</math>. A point <math>P_i</math> is called "peripheral" if, f :'''Input:''' real numbers <math>U < 1</math>; ...

* [[Division (mathematics)|divide]] their weight by the [[exponentiation|square]] of their height ...to help decide whether people are too fat or too thin. The WHO uses these numbers for adults:<ref>{{cite web|url = http://www.who.int/bmi/index.jsp?introPage ...

||[[Square root]] of 5, [[Gauss]] sum ||<small>The ratio of a square and circumscribed or inscribed circles</small> ...

...of a real-valued argument ''x''. (This means both the input and output are numbers.) ...r, we can restrict both its domain and codomain to the set of non-negative numbers (0,+∞) to get an (invertible) bijection (see examples below). ...

* Step 1: Numbers can be [[Primality test#Probabilistic tests|probabilistically tested]] for Since <math>p\,</math> and <math>q\,</math> are distinct prime numbers, applying the [[Chinese remainder theorem]] to these two congruences yields ...

...or fixed-point arithmetic; when computing mathematical functions such as [[square root]]s, [[logarithm]]s, and [[sine]]s; or when using a [[floating point]] ...t occurs when [[physical quantity|physical quantities]] must be encoded by numbers or [[digital signal]]s. ...

<math>\square</math> ...to introduce the concepts of the <math>k</math>-cascade representation of numbers and the colex order of sets. ...

<math>\square</math> ...to introduce the concepts of the <math>k</math>-cascade representation of numbers and the colex order of sets. ...

:For <math>n>0</math>, the numbers of subsets of an <math>n</math>-set of even and of odd cardinality are equa For counting problems, what we care about are ''numbers''. In the binomial theorem, a formal ''variable'' <math>x</math> is introdu ...

:For <math>n>0</math>, the numbers of subsets of an <math>n</math>-set of even and of odd cardinality are equa For counting problems, what we care about are ''numbers''. In the binomial theorem, a formal ''variable'' <math>x</math> is introdu ...

:For <math>n>0</math>, the numbers of subsets of an <math>n</math>-set of even and of odd cardinality are equa For counting problems, what we care about are ''numbers''. In the binomial theorem, a formal ''variable'' <math>x</math> is introdu ...

:For <math>n>0</math>, the numbers of subsets of an <math>n</math>-set of even and of odd cardinality are equa For counting problems, what we care about are ''numbers''. In the binomial theorem, a formal ''variable'' <math>x</math> is introdu ...

:For <math>n>0</math>, the numbers of subsets of an <math>n</math>-set of even and of odd cardinality are equa For counting problems, what we care about are ''numbers''. In the binomial theorem, a formal ''variable'' <math>x</math> is introdu ...

:For <math>n>0</math>, the numbers of subsets of an <math>n</math>-set of even and of odd cardinality are equa For counting problems, what we care about are ''numbers''. In the binomial theorem, a formal ''variable'' <math>x</math> is introdu ...

:For <math>n>0</math>, the numbers of subsets of an <math>n</math>-set of even and of odd cardinality are equa For counting problems, what we care about are ''numbers''. In the binomial theorem, a formal ''variable'' <math>x</math> is introdu ...

:For <math>n>0</math>, the numbers of subsets of an <math>n</math>-set of even and of odd cardinality are equa For counting problems, what we care about are ''numbers''. In the binomial theorem, a formal ''variable'' <math>x</math> is introdu ...

:For <math>n>0</math>, the numbers of subsets of an <math>n</math>-set of even and of odd cardinality are equa For counting problems, what we care about are ''numbers''. In the binomial theorem, a formal ''variable'' <math>x</math> is introdu ...

...input a set <math>S</math> of <math>n</math> numbers, we want to sort the numbers in <math>S</math> in increasing order. One of the most famous algorithm for ...re all numbers in <math>S_1</math> are smaller than <math>x</math> and all numbers in <math>S_2</math> are larger than <math>x</math>; ...

...v</sub> || E 2C₄ C₂ 2σ_v 2σ_d || square pyramidal || [[xenon oxytetrafluoride]] ...ub>'' ''i'' 2S₄ σ_h 2σ_v 2σ_d || square planar || [[xenon tetrafluoride]] ...

If there exists a '''[[square matrix]]''' called '''''A''''', a '''[[scalar]] ''λ''''', and a '''non-zero ...cation. For example, instead of [[real numbers]], scalars may be [[complex numbers]]; instead of arrows, vectors may be [[function (mathematics)|functions]] ...

Sampling <math>t</math> mutually independent random numbers from <math>[p]</math> can be quite expensive since it requires <math>\Omega The error is reduced to <math>1/t</math> with only two random numbers. This scheme works as long as <math>t\le p</math>. ...

Sampling <math>t</math> mutually independent random numbers from <math>[p]</math> can be quite expensive since it requires <math>\Omega The error is reduced to <math>1/t</math> with only two random numbers. This scheme works as long as <math>t\le p</math>. ...

...ximation ratio of 2 to being 2-universal. The proof uses the fact that odd numbers are relative prime to a power of 2. We exploit that C-multiplication (*) of unsigned u-bit numbers is done <math>\bmod 2^u</math>, and have a one-line C-code for computing th ...

...ximation ratio of 2 to being 2-universal. The proof uses the fact that odd numbers are relative prime to a power of 2. We exploit that C-multiplication (*) of unsigned u-bit numbers is done <math>\bmod 2^u</math>, and have a one-line C-code for computing th ...

Sampling <math>t</math> mutually independent random numbers from <math>[p]</math> can be quite expensive since it requires <math>\Omega The error is reduced to <math>1/t</math> with only two random numbers. This scheme works as long as <math>t\le p</math>. ...

<math>\square</math> ...to introduce the concepts of the <math>k</math>-cascade representation of numbers and the colex order of sets. ...

...ted as a diagram of dots (or squares), called a '''Ferrers diagram''' (the square version of Ferrers diagram is also called a '''Young diagram''', named afte |align=center|Ferrers diagram (''square version'') of (5,4,2,1) ...

...re related. For a vector '''a''', the dot product '''a''' · '''a''' is the square of the length of '''a''', or Unlike multiplication of ordinary numbers, where if ''ab'' = ''ac'', then ''b'' always equals ''c'' unless ''a'' is z ...

...ximation ratio of 2 to being 2-universal. The proof uses the fact that odd numbers are relative prime to a power of 2. We exploit that C-multiplication (*) of unsigned u-bit numbers is done <math>\bmod 2^u</math>, and have a one-line C-code for computing th ...

...multiply amplitude by amplitude (or in other words he had to [[wikt:square|square]] the amplitude) to get the intensity he wanted. He had to figure out how t ...no problem. But they were both complicated equations, and how you got the numbers to plug into the equations turned out to be different depending on which wa ...

...nd therefore frequency). So the formula that Planck worked out to make the numbers all come out right was: Putting in different numbers for '''m''' and '''n''', it is easy to predict frequencies for many types o ...

...ximation ratio of 2 to being 2-universal. The proof uses the fact that odd numbers are relative prime to a power of 2. We exploit that C-multiplication (*) of unsigned u-bit numbers is done <math>\bmod 2^u</math>, and have a one-line C-code for computing th ...