Search results
Results from the WOW.Com Content Network
Square roots of negative numbers are called imaginary because in early-modern mathematics, only what are now called real numbers, obtainable by physical measurements or basic arithmetic, were considered to be numbers at all – even negative numbers were treated with skepticism – so the square root of a negative number was previously considered undefined or nonsensical.
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]
A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the "imaginary unit", that satisfies i 2 = −1.
The imaginary unit's core property is that i 2 = −1. The term "imaginary" was coined because there is no number having a negative square. There are in fact two complex square roots of −1, namely i and −i, just as there are two complex square roots of every other real number (except zero, which has one double square root).
Quadratic surd: A root of a quadratic equation with rational coefficients. Such a number is algebraic and can be expressed as the sum of a rational number and the square root of a rational number. Constructible number: A number representing a length that can be constructed using a compass and straightedge.
The square root of a positive integer is the product of the roots of its prime factors, because the square root of a product is the product of the square roots of the factors. Since p 2 k = p k , {\textstyle {\sqrt {p^{2k}}}=p^{k},} only roots of those primes having an odd power in the factorization are necessary.
is the imaginary unit, which by definition satisfies =, and π {\displaystyle \pi } is pi , the ratio of the circumference of a circle to its diameter . Euler's identity is named after the Swiss mathematician Leonhard Euler .
So one continuous motion in the complex plane has transformed the positive square root e 0 = 1 into the negative square root e iπ = −1. This problem arises because the point z = 0 has just one square root, while every other complex number z ≠ 0 has exactly two square roots.