Imaginary unit — Safekipedia Discoverer

The imaginary unit, usually denoted by i, is a special mathematical constant that helps solve problems that cannot be solved with regular numbers. It is defined as the solution to the quadratic equation x² = −1, which has no answer among the real numbers. This means that when you square the imaginary unit (multiply it by itself), you get -1, which is not possible with ordinary numbers.

When we combine real numbers with the imaginary unit using addition and multiplication, we create a new and powerful number system called complex numbers. Complex numbers look like a + bi, where a and b are real numbers. This system is widely used in many areas of science and engineering because it provides a complete way to solve all kinds of equations.

There are two complex square roots of −1: the imaginary unit i and its opposite, −i. In some fields, such as electrical engineering, the letter j is used instead of i to avoid confusion with electric current. This shows how important and useful the imaginary unit is in solving real-world problems.

Terminology

Further information: Complex number § History

Square roots of negative numbers are called imaginary because long ago, only numbers that could be measured in the real world were thought to be true numbers. Even negative numbers were viewed with doubt, so finding the square root of a negative number seemed impossible and made up. The word imaginary was first used by René Descartes, and Isaac Newton used it as early as 1670. The symbol i for this special number was introduced by Leonhard Euler.

Definition

The imaginary unit, often written as i, is a special number in math. It is defined by the rule that i × i = –1. This means that i is a solution to the equation x² = –1, which has no solution among real numbers.

Because of this rule, we can use i to create new numbers called complex numbers. A complex number looks like a + bi, where a and b are real numbers. The i part lets us work with numbers that have both a real side and an imaginary side. This helps solve many problems in math, science, and engineering.

The powers of i are cyclic:
⋮ {\displaystyle \ \vdots }
i − 4 = − 1 i {\displaystyle \ i^{-4}={\phantom {-}}1{\phantom {i}}}
i − 3 = − i 1 {\displaystyle \ i^{-3}={\phantom {-}}i{\phantom {1}}}
i − 2 = − 1 i {\displaystyle \ i^{-2}=-1{\phantom {i}}}
i − 1 = − i 1 {\displaystyle \ i^{-1}=-i{\phantom {1}}}
i 0 = − 1 i {\displaystyle \ \ i^{0}\ ={\phantom {-}}1{\phantom {i}}}
i 1 = − i 1 {\displaystyle \ \ i^{1}\ ={\phantom {-}}i{\phantom {1}}}
i 2 = − 1 i {\displaystyle \ \ i^{2}\ =-1{\phantom {i}}}
i 3 = − i 1 {\displaystyle \ \ i^{3}\ =-i{\phantom {1}}}
i 4 = − 1 i {\displaystyle \ \ i^{4}\ ={\phantom {-}}1{\phantom {i}}}
i 5 = − i 1 {\displaystyle \ \ i^{5}\ ={\phantom {-}}i{\phantom {1}}}
i 6 = − 1 i {\displaystyle \ \ i^{6}\ =-1{\phantom {i}}}
i 7 = − i 1 {\displaystyle \ \ i^{7}\ =-i{\phantom {1}}}
⋮ {\displaystyle \ \vdots }

Proper use

The imaginary unit is often shown as the square root of -1. However, we must be very careful when working with these kinds of expressions. The symbol for a square root is usually only used for finding the positive square root of a positive real number. If we try to use the same rules for square roots of complex numbers, we can get wrong answers. For example, one might incorrectly think that the square root of -1 times the square root of -1 equals the square root of 1, which is 1, but this is not true.

To avoid mistakes, especially when dealing with negative numbers, it is better to write expressions like i times the square root of 7, rather than the square root of -7. For more information, see the articles on Square root and Branch point.

Properties

The imaginary unit, denoted by i, is a special number in math that helps solve equations that no real number can solve. For example, it is the solution to the equation x² = −1. When we combine real numbers with i, we create complex numbers, which look like a + bi, where a and b are real numbers.

Imaginary numbers follow special rules. Adding or multiplying them creates patterns, and they can be visualized on a number line called the imaginary axis. The imaginary unit i has interesting properties, like rotating numbers in a circle when multiplied, and its powers repeat in a cycle every four steps. These ideas help mathematicians solve many kinds of problems in fields like engineering, physics, and computer science.

Main article: Complex arithmetic