Imaginary unit — Safekipedia Adventurer

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.

When we combine real numbers with the imaginary unit using addition and multiplication, we create a new number system called complex numbers. Complex numbers look like a + bi, where a and b are real numbers. This system is used in many areas of science and engineering because it helps solve many different 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.

Terminology

Further information: Complex number § History

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

Definition

The imaginary unit, written as i, is a special number used in math. It follows one simple rule: i × i = –1. This means i helps solve the equation x² = –1, which cannot be solved with regular numbers we use every day.

With i, we can make new numbers called complex numbers. A complex number looks like a + bi, where a and b are real numbers. The i part lets us mix real numbers with imaginary numbers. This is useful for solving 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 shown as the square root of -1. But we need to be careful with this. The symbol for a square root is usually for finding the positive square root of a positive real number. If we use the same rules for complex numbers, we can get wrong answers. For example, someone might 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 correct.

To avoid mistakes, especially with negative numbers, it is better to write expressions like i times the square root of 7, instead of 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