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The square of the absolute value of a complex number is called its absolute square, squared modulus, or squared magnitude. [1] [better source needed] It is the product of the complex number with its complex conjugate, and equals the sum of the squares of the real and imaginary parts of the complex number.
Gauss [10] pointed out that the four squares theorem follows easily from the fact that any positive integer that is 1 or 2 mod 4 is a sum of 3 squares, because any positive integer not divisible by 4 can be reduced to this form by subtracting 0 or 1 from it. However, proving the three-square theorem is considerably more difficult than a direct ...
For example, 9 is a square number, since it equals 3 2 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 2, usually pronounced as "n squared". The name square number comes from the name of the shape.
It is the second Fibonacci prime (and the second Lucas prime), the second Sophie Germain prime, the third Harshad number in base 10, and the second factorial prime, as it is equal to 2! + 1. 3 is the second and only prime triangular number, [5] and Gauss proved that every integer is the sum of at most 3 triangular numbers.
3.6 × 10 −4951 is approximately equal to the smallest non-zero value that can be represented by an 80-bit x86 double-extended IEEE floating-point value. 1 × 10 −398 is equal to the smallest non-zero value that can be represented by a double-precision IEEE decimal floating-point value.
The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3. It is denoted mathematically as 3 {\textstyle {\sqrt {3}}} or 3 1 / 2 {\displaystyle 3^{1/2}} . It is more precisely called the principal square root of 3 to distinguish it from the negative number with the same property.
where C is the circumference of a circle, d is the diameter, and r is the radius.More generally, = where L and w are, respectively, the perimeter and the width of any curve of constant width.
On the other hand, the primes 3, 7, 11, 19, 23 and 31 are all congruent to 3 modulo 4, and none of them can be expressed as the sum of two squares. This is the easier part of the theorem, and follows immediately from the observation that all squares are congruent to 0 (if number squared is even) or 1 (if number squared is odd) modulo 4.