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Every real number squared gives a positive answer or zero, which makes an equation like x2+1=0x^2 + 1 = 0 seem completely impossible to solve.
For centuries, mathematicians hit a brick wall whenever a calculation produced the square root of a negative number — until they decided to invent a brand new kind of number to handle it.

The Imaginary Unit

In 1572, Italian mathematician Rafael Bombelli developed the first formal rules for working with square roots of negative numbers while solving cubic equations, and Leonhard Euler later introduced the symbol ii for this concept.
We define the imaginary unit ii as the principal square root of −1-1, meaning i=−1i = \sqrt{-1} and i2=−1i^2 = -1.
Using the product rule for radicals, we can rewrite any negative square root in terms of ii, such as −16=16×−1=4i\sqrt{-16} = \sqrt{16} \times \sqrt{-1} = 4i.
What happens when you combine an ordinary real number with an imaginary one? You create a complex number.

The Anatomy of a Complex Number

A complex number is any number that can be written in the standard form z=a+biz = a + bi, where aa and bb are both real numbers.
The term aa is called the real part of zz, denoted as Re(z)\text{Re}(z), while bb is called the imaginary part, denoted as Im(z)\text{Im}(z).
Let's inspect how real and imaginary parts work across different types of numbers.