Complex number

A complex number is a number, but is different from common numbers in many ways. A complex number is made up using two numbers combined. The first part is a real number, and the second part is an imaginary number. The most important imaginary number is called ${\displaystyle i}$, defined as a number that will be -1 when squared ("squared" means "multiplied by itself"): ${\displaystyle i^{2}=i\times i=-1\ }$. All the other imaginary numbers are ${\displaystyle i}$ multiplied by a real number, in the same way that all real numbers can be thought of as 1 multiplied by another number. Arithmetic functions such as addition, subtraction, multiplication, and division can be used with complex numbers. They also follow commutative, associative and distributive properties, just like real numbers. The set of complex numbers is often represented using the symbol ${\displaystyle \mathbb {C} }$.^[1][2]

Complex numbers were discovered while attempting to solve special equations that have exponents in them. These began to pose real problems for mathematicians. As a comparison, using negative numbers, it is possible to find the x in the equation ${\displaystyle a+x=b}$ for all real values of a and b, but if only positive numbers are allowed for x, it is sometimes impossible to find a positive x, as in the equation ${\displaystyle 3+x=1}$.

With exponentiation, there is a difficulty to be overcome.^[3] There is no real number that gives −1 when it is squared. In other words, −1 (or any other negative number) has no real square root. For example, there is no real number ${\displaystyle x}$ that solves the equation ${\displaystyle (x+1)^{2}=-9}$. To solve this problem, mathematicians introduced a symbol i and called it the imaginary unit.^[1] This is the imaginary number that will give −1 when it is squared.

The first mathematicians to have thought of this were probably Gerolamo Cardano and Rafael Bombelli. They lived in the 16th century.^[2] It was probably Leonhard Euler who introduced writing ${\displaystyle \mathrm {i} }$ for that number.

All complex numbers can be written as ${\displaystyle a+bi}$^[3] (or ${\displaystyle a+b\cdot i}$), where a is called the real part of the number, and b is called the imaginary part. We write ${\displaystyle \Re (z)}$ or ${\displaystyle \operatorname {Re} (z)}$ for the real part of a complex number ${\displaystyle z}$. So, if ${\displaystyle z=a+bi}$, we write ${\displaystyle a=\Re (z)=\operatorname {Re} (z)}$. Similarly, we write ${\displaystyle \Im (z)}$ or ${\displaystyle \operatorname {Im} (z)}$ for the imaginary part of a complex number ${\displaystyle z}$; ${\displaystyle b=\Im (z)=\operatorname {Im} (z)}$, for the same z.^[1] Every real number is also a complex number; it is a complex number z with ${\displaystyle \Im (z)=0}$.

A complex number can also be written as an ordered pair ${\displaystyle (a,b)}$, where both a and b are real numbers. Any real number can simply be written as ${\displaystyle a+0\cdot i}$, or as the pair ${\displaystyle (a,0)}$.^[3]

Sometimes, ${\displaystyle j}$ is written instead of ${\displaystyle i}$. In electrical engineering for instance, ${\displaystyle i}$ means electric current, so writing ${\displaystyle i}$ can cause a lot of problems because some numbers in electrical engineering are complex numbers.

1. ↑ ^{1.0 1.1 1.2} "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-08-12.
2. ↑ ^{2.0 2.1} "Complex Numbers | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-08-12.
3. ↑ ^{3.0 3.1 3.2} Weisstein, Eric. "Complex numbers". Wolfram MathWorld. Retrieved August 11, 2020.