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For example, the equation (+) = has no real solution, because the square of a real number cannot be negative, but has the two nonreal complex solutions + and . Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule i 2 = − 1 {\displaystyle i^{2}=-1} along with the associative , commutative , and ...
The imaginary unit i in the complex plane: Real numbers are conventionally drawn on the horizontal axis, and imaginary numbers on the vertical axis.. The imaginary unit or unit imaginary number (i) is a mathematical constant that is a solution to the quadratic equation x 2 + 1 = 0.
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.
This formula can be interpreted as saying that the function e iφ is a unit complex number, i.e., it traces out the unit circle in the complex plane as φ ranges through the real numbers. Here φ is the angle that a line connecting the origin with a point on the unit circle makes with the positive real axis , measured counterclockwise and in ...
An imaginary number is the product of a real number and the imaginary unit i, [note 1] which is defined by its property i 2 = −1. [1] [2] The square of an imaginary number bi is −b 2. For example, 5i is an imaginary number, and its square is −25. The number zero is considered to be both real and imaginary. [3]
Square roots of negative numbers can be discussed within the framework of complex numbers. More generally, square roots can be considered in any context in which a notion of the "square" of a mathematical object is defined. These include function spaces and square matrices, among other mathematical structures.
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers, then the complex conjugate of a + b i {\displaystyle a+bi} is a − b i . {\displaystyle a-bi.}
The difference of two squares is used to find the linear factors of the sum of two squares, using complex number coefficients. For example, the complex roots of z 2 + 4 {\displaystyle z^{2}+4} can be found using difference of two squares:
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