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54-461: 36 is both the square of six , and the eighth triangular number or the sum of the first eight non-zero positive integers , which makes 36 the first non-trivial square triangular number . Aside from being the smallest square triangular number other than 1 , it is also the only triangular number (other than 1) whose square root is also a triangular number. 36 is also the eighth refactorable number , as it has exactly nine positive divisors, and 9

108-432: A + b ) ( a − b ) {\displaystyle a^{2}-b^{2}=(a+b)(a-b)} This is the difference-of-squares formula , which can be useful for mental arithmetic: for example, 47 × 53 can be easily computed as 50 − 3 = 2500 − 9 = 2491 . A square number is also the sum of two consecutive triangular numbers . The sum of two consecutive square numbers is a centered square number . Every odd square

162-464: A curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept). Two different regions may have the same area (as in squaring the circle ); by synecdoche , "area" sometimes is used to refer to the region, as in a " polygonal area ". The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI),

216-400: A definite integral : The formula for the area enclosed by an ellipse is related to the formula of a circle; for an ellipse with semi-major and semi-minor axes x and y the formula is: Most basic formulas for surface area can be obtained by cutting surfaces and flattening them out (see: developable surfaces ). For example, if the side surface of a cylinder (or any prism )

270-531: A perfect number . The sum of the n first square numbers is ∑ n = 0 N n 2 = 0 2 + 1 2 + 2 2 + 3 2 + 4 2 + ⋯ + N 2 = N ( N + 1 ) ( 2 N + 1 ) 6 . {\displaystyle \sum _{n=0}^{N}n^{2}=0^{2}+1^{2}+2^{2}+3^{2}+4^{2}+\cdots +N^{2}={\frac {N(N+1)(2N+1)}{6}}.} The first values of these sums,

324-450: A prime number has factors of only 1 and itself, and since m = 2 is the only non-zero value of m to give a factor of 1 on the right side of the equation above, it follows that 3 is the only prime number one less than a square ( 3 = 2 − 1 ). More generally, the difference of the squares of two numbers is the product of their sum and their difference. That is, a 2 − b 2 = (

378-403: A shape or planar lamina , while surface area refers to the area of an open surface or the boundary of a three-dimensional object . Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of

432-411: A 4×4 checkerboard is 36. Since it is possible to find sequences of 36 consecutive integers such that each inner member shares a factor with either the first or the last member, 36 is an Erdős–Woods number . The sum of the integers from 1 to 36 is 666 (see number of the beast ). 36 is also a Tridecagonal number. Square number In mathematics , a square number or perfect square

486-417: A corresponding unit of area, namely the area of a square with the given side length. Thus areas can be measured in square metres (m ), square centimetres (cm ), square millimetres (mm ), square kilometres (km ), square feet (ft ), square yards (yd ), square miles (mi ), and so forth. Algebraically, these units can be thought of as the squares of the corresponding length units. The SI unit of area

540-428: A rectangle with length l and width w , the formula for the area is: That is, the area of the rectangle is the length multiplied by the width. As a special case, as l = w in the case of a square, the area of a square with side length s is given by the formula: The formula for the area of a rectangle follows directly from the basic properties of area, and is sometimes taken as a definition or axiom . On

594-433: A solid shape such as a sphere , cone, or cylinder, the area of its boundary surface is called the surface area . Formulas for the surface areas of simple shapes were computed by the ancient Greeks , but computing the surface area of a more complicated shape usually requires multivariable calculus . Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area

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648-418: A square number can end only with square digits (like in base 12, a prime number can end only with prime digits or 1), that is, 0, 1, 4 or 9, as follows: Similar rules can be given for other bases, or for earlier digits (the tens instead of the units digit, for example). All such rules can be proved by checking a fixed number of cases and using modular arithmetic . In general, if a prime   p divides

702-642: A square number  m then the square of p must also divide m ; if p fails to divide ⁠ m / p ⁠ , then m is definitely not square. Repeating the divisions of the previous sentence, one concludes that every prime must divide a given perfect square an even number of times (including possibly 0 times). Thus, the number m is a square number if and only if, in its canonical representation , all exponents are even. Squarity testing can be used as alternative way in factorization of large numbers. Instead of testing for divisibility, test for squarity: for given m and some number  k , if k − m

756-404: A triangular number minus eight. Since all triangular numbers have an odd factor, but no two values of 2 differ by an amount containing an odd factor, the only perfect square of the form 2 − 1 is 1, and the only perfect square of the form 2 + 1 is 9. Area Area is the measure of a region 's size on a surface . The area of a plane region or plane area refers to the area of

810-630: Is n , with 0 = 0 being the zeroth one. The concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two square integers, and, conversely, the ratio of two square integers is a square, for example, 4 9 = ( 2 3 ) 2 {\displaystyle \textstyle {\frac {4}{9}}=\left({\frac {2}{3}}\right)^{2}} . Starting with 1, there are ⌊ m ⌋ {\displaystyle \lfloor {\sqrt {m}}\rfloor } square numbers up to and including m , where

864-422: Is also a centered octagonal number . Another property of a square number is that (except 0) it has an odd number of positive divisors, while other natural numbers have an even number of positive divisors. An integer root is the only divisor that pairs up with itself to yield the square number, while other divisors come in pairs. Lagrange's four-square theorem states that any positive integer can be written as

918-409: Is always even. In other words, all odd square numbers have a remainder of 1 when divided by 8. Every odd perfect square is a centered octagonal number . The difference between any two odd perfect squares is a multiple of 8. The difference between 1 and any higher odd perfect square always is eight times a triangular number, while the difference between 9 and any higher odd perfect square is eight times

972-459: Is always the product of m − 1 {\displaystyle m-1} and m + 1 ; {\displaystyle m+1;} that is, m 2 − 1 = ( m − 1 ) ( m + 1 ) . {\displaystyle m^{2}-1=(m-1)(m+1).} For example, since 7 = 49 , one has 6 × 8 = 48 {\displaystyle 6\times 8=48} . Since

1026-405: Is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it equals 3 and can be written as 3 × 3 . The usual notation for the square of a number n is not the product n  ×  n , but the equivalent exponentiation n , usually pronounced as " n squared". The name square number comes from

1080-447: Is approximately triangular in shape, and the sectors can be rearranged to form an approximate parallelogram. The height of this parallelogram is r , and the width is half the circumference of the circle, or π r . Thus, the total area of the circle is π r : Though the dissection used in this formula is only approximate, the error becomes smaller and smaller as the circle is partitioned into more and more sectors. The limit of

1134-418: Is cut lengthwise, the surface can be flattened out into a rectangle. Similarly, if a cut is made along the side of a cone , the side surface can be flattened out into a sector of a circle, and the resulting area computed. The formula for the surface area of a sphere is more difficult to derive: because a sphere has nonzero Gaussian curvature , it cannot be flattened out. The formula for the surface area of

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1188-462: Is known as Heron's formula for the area of a triangle in terms of its sides, and a proof can be found in his book, Metrica , written around 60 CE. It has been suggested that Archimedes knew the formula over two centuries earlier, and since Metrica is a collection of the mathematical knowledge available in the ancient world, it is possible that the formula predates the reference given in that work. In 300 BCE Greek mathematician Euclid proved that

1242-463: Is one of them; in fact, it is the smallest positive integer with at least nine divisors , which leads 36 to be the 7th highly composite number . It is the sum of the fourth pair of twin-primes ( 17 + 19 ), and the 18th Harshad number in decimal , as it is divisible by the sum of its digits (9). It is the smallest number n {\displaystyle n} with exactly eight solutions ( 37 , 57 , 63 , 74 , 76 , 108 , 114 , 126 ) to

1296-431: Is related to the definition of determinants in linear algebra , and is a basic property of surfaces in differential geometry . In analysis , the area of a subset of the plane is defined using Lebesgue measure , though not every subset is measurable if one supposes the axiom of choice. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions. Area can be defined through

1350-470: Is the square metre, which is considered an SI derived unit . Calculation of the area of a square whose length and width are 1 metre would be: 1 metre × 1 metre = 1 m and so, a rectangle with different sides (say length of 3 metres and width of 2 metres) would have an area in square units that can be calculated as: 3 metres × 2 metres = 6 m . This is equivalent to 6 million square millimetres. Other useful conversions are: In non-metric units,

1404-491: Is the square of an integer  n then k − n divides m . (This is an application of the factorization of a difference of two squares .) For example, 100 − 9991 is the square of 3, so consequently 100 − 3 divides 9991. This test is deterministic for odd divisors in the range from k − n to k + n where k covers some range of natural numbers k ≥ m . {\displaystyle k\geq {\sqrt {m}}.} A square number cannot be

1458-455: The Cartesian coordinates ( x i , y i ) {\displaystyle (x_{i},y_{i})} ( i =0, 1, ..., n -1) of whose n vertices are known, the area is given by the surveyor's formula : where when i = n -1, then i +1 is expressed as modulus n and so refers to 0. The most basic area formula is the formula for the area of a rectangle . Given

1512-470: The Euler totient function ϕ ( x ) = n {\displaystyle \phi (x)=n} . Adding up some subsets of its divisors (e.g., 6, 12, and 18) gives 36; hence, it is also the eighth semiperfect number . This number is the sum of the cubes of the first three positive integers and also the product of the squares of the first three positive integers. 36 is the number of degrees in

1566-478: The hectare is still commonly used to measure land: Other uncommon metric units of area include the tetrad , the hectad , and the myriad . The acre is also commonly used to measure land areas, where An acre is approximately 40% of a hectare. On the atomic scale, area is measured in units of barns , such that: The barn is commonly used in describing the cross-sectional area of interaction in nuclear physics . In South Asia (mainly Indians), although

1620-498: The interior angle of each tip of a regular pentagram . The thirty-six officers problem is a mathematical puzzle with no solution . The number of possible outcomes (not summed) in the roll of two distinct dice . 36 is the largest numeric base that some computer systems support because it exhausts the numerals, 0–9, and the letters, A-Z. See Base 36 . The truncated cube and the truncated octahedron are Archimedean solids with 36 edges. The number of domino tilings of

1674-404: The n th square number can be computed from the previous square by n = ( n − 1) + ( n − 1) + n = ( n − 1) + (2 n − 1) . Alternatively, the n th square number can be calculated from the previous two by doubling the ( n  − 1) th square, subtracting the ( n  − 2) th square number, and adding 2, because n = 2( n − 1) − ( n − 2) + 2 . For example, The square minus one of a number m

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1728-409: The real number system , square numbers are non-negative . A non-negative integer is a square number when its square root is again an integer. For example, 9 = 3 , {\displaystyle {\sqrt {9}}=3,} so 9 is a square number. A positive integer that has no square divisors except 1 is called square-free . For a non-negative integer n , the n th square number

1782-485: The square pyramidal numbers , are: (sequence A000330 in the OEIS ) 0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201... The sum of the first odd integers, beginning with one, is a perfect square: 1, 1 + 3, 1 + 3 + 5, 1 + 3 + 5 + 7, etc. This explains Galileo's law of odd numbers : if a body falling from rest covers one unit of distance in

1836-417: The surveyor's formula for the area of any polygon with known vertex locations by Gauss in the 19th century. The development of integral calculus in the late 17th century provided tools that could subsequently be used for computing more complicated areas, such as the area of an ellipse and the surface areas of various curved three-dimensional objects. For a non-self-intersecting ( simple ) polygon,

1890-556: The 5th century BCE, Hippocrates of Chios was the first to show that the area of a disk (the region enclosed by a circle) is proportional to the square of its diameter, as part of his quadrature of the lune of Hippocrates , but did not identify the constant of proportionality . Eudoxus of Cnidus , also in the 5th century BCE, also found that the area of a disk is proportional to its radius squared. Subsequently, Book I of Euclid's Elements dealt with equality of areas between two-dimensional figures. The mathematician Archimedes used

1944-466: The area of a cyclic quadrilateral (a quadrilateral inscribed in a circle) in terms of its sides. In 1842, the German mathematicians Carl Anton Bretschneider and Karl Georg Christian von Staudt independently found a formula, known as Bretschneider's formula , for the area of any quadrilateral. The development of Cartesian coordinates by René Descartes in the 17th century allowed the development of

1998-501: The area of a triangle is half that of a parallelogram with the same base and height in his book Elements of Geometry . In 499 Aryabhata , a great mathematician - astronomer from the classical age of Indian mathematics and Indian astronomy , expressed the area of a triangle as one-half the base times the height in the Aryabhatiya . In the 7th century CE, Brahmagupta developed a formula, now known as Brahmagupta's formula , for

2052-405: The areas of simple shapes such as triangles , rectangles , and circles . Using these formulas, the area of any polygon can be found by dividing the polygon into triangles . For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus . For

2106-417: The areas of the approximate parallelograms is exactly π r , which is the area of the circle. This argument is actually a simple application of the ideas of calculus . In ancient times, the method of exhaustion was used in a similar way to find the area of the circle, and this method is now recognized as a precursor to integral calculus . Using modern methods, the area of a circle can be computed using

2160-417: The conversion between two square units is the square of the conversion between the corresponding length units. the relationship between square feet and square inches is where 144 = 12 = 12 × 12. Similarly: In addition, conversion factors include: There are several other common units for area. The are was the original unit of area in the metric system , with: Though the are has fallen out of use,

2214-520: The countries use SI units as official, many South Asians still use traditional units. Each administrative division has its own area unit, some of them have same names, but with different values. There's no official consensus about the traditional units values. Thus, the conversions between the SI units and the traditional units may have different results, depending on what reference that has been used. Some traditional South Asian units that have fixed value: In

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2268-432: The expression ⌊ x ⌋ {\displaystyle \lfloor x\rfloor } represents the floor of the number  x . The squares (sequence A000290 in the OEIS ) smaller than 60  = 3600 are: The difference between any perfect square and its predecessor is given by the identity n − ( n − 1) = 2 n − 1 . Equivalently, it is possible to count square numbers by adding together

2322-413: The first arbitrary time interval, it covers 3, 5, 7, etc., units of distance in subsequent time intervals of the same length. From s = u t + 1 2 a t 2 {\displaystyle s=ut+{\tfrac {1}{2}}at^{2}} , for u = 0 and constant a (acceleration due to gravity without air resistance); so s is proportional to t , and the distance from

2376-780: The last square, the last square's root, and the current root, that is, n = ( n − 1) + ( n − 1) + n . The number m is a square number if and only if one can arrange m points in a square: The expression for the n th square number is n . This is also equal to the sum of the first n odd numbers as can be seen in the above pictures, where a square results from the previous one by adding an odd number of points (shown in magenta). The formula follows: n 2 = ∑ k = 1 n ( 2 k − 1 ) . {\displaystyle n^{2}=\sum _{k=1}^{n}(2k-1).} For example, 5 = 25 = 1 + 3 + 5 + 7 + 9 . There are several recursive methods for computing square numbers. For example,

2430-406: The left. If the triangle is moved to the other side of the trapezoid, then the resulting figure is a rectangle. It follows that the area of the parallelogram is the same as the area of the rectangle: However, the same parallelogram can also be cut along a diagonal into two congruent triangles, as shown in the figure to the right. It follows that the area of each triangle is half the area of

2484-443: The name of the shape. The unit of area is defined as the area of a unit square ( 1 × 1 ). Hence, a square with side length n has area n . If a square number is represented by n points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of n ; thus, square numbers are a type of figurate numbers (other examples being cube numbers and triangular numbers ). In

2538-432: The other hand, if geometry is developed before arithmetic , this formula can be used to define multiplication of real numbers . Most other simple formulas for area follow from the method of dissection . This involves cutting a shape into pieces, whose areas must sum to the area of the original shape. For an example, any parallelogram can be subdivided into a trapezoid and a right triangle , as shown in figure to

2592-421: The parallelogram: Similar arguments can be used to find area formulas for the trapezoid as well as more complicated polygons . The formula for the area of a circle (more properly called the area enclosed by a circle or the area of a disk ) is based on a similar method. Given a circle of radius r , it is possible to partition the circle into sectors , as shown in the figure to the right. Each sector

2646-413: The standard unit of area is the square metre (written as m ), which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics , the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number . There are several well-known formulas for

2700-844: The starting point are consecutive squares for integer values of time elapsed. The sum of the n first cubes is the square of the sum of the n first positive integers; this is Nicomachus's theorem . All fourth powers, sixth powers, eighth powers and so on are perfect squares. A unique relationship with triangular numbers T n {\displaystyle T_{n}} is: ( T n ) 2 + ( T n + 1 ) 2 = T ( n + 1 ) 2 {\displaystyle (T_{n})^{2}+(T_{n+1})^{2}=T_{(n+1)^{2}}} Squares of even numbers are even, and are divisible by 4, since (2 n ) = 4 n . Squares of odd numbers are odd, and are congruent to 1 modulo 8, since (2 n + 1) = 4 n ( n + 1) + 1 , and n ( n + 1)

2754-420: The sum of four or fewer perfect squares. Three squares are not sufficient for numbers of the form 4 (8 m + 7) . A positive integer can be represented as a sum of two squares precisely if its prime factorization contains no odd powers of primes of the form 4 k + 3 . This is generalized by Waring's problem . In base 10 , a square number can end only with digits 0, 1, 4, 5, 6 or 9, as follows: In base 12 ,

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2808-410: The tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, in his book Measurement of a Circle . (The circumference is 2 π r , and the area of a triangle is half the base times the height, yielding the area π r for the disk.) Archimedes approximated

2862-511: The use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists. An approach to defining what is meant by "area" is through axioms . "Area" can be defined as a function from a collection M of a special kinds of plane figures (termed measurable sets) to the set of real numbers, which satisfies the following properties: It can be proved that such an area function actually exists. Every unit of length has

2916-419: The value of π (and hence the area of a unit-radius circle) with his doubling method , in which he inscribed a regular triangle in a circle and noted its area, then doubled the number of sides to give a regular hexagon , then repeatedly doubled the number of sides as the polygon's area got closer and closer to that of the circle (and did the same with circumscribed polygons ). Heron of Alexandria found what

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