# Complex Numbers

Complex Numbers have a real component and an imaginary component. In the real number system it is not possible to take the square root of a negative number. For example, it is not possible to simplify $\sqrt{-9}$ because there is not a number that when squared will equal -9. However, in the set of complex numbers it is possible to take the square root of a negative number by defining $\sqrt{-1}$ as i an imaginary number.

For Example

• $\sqrt{-16}$ can be written as $\sqrt{16}x\sqrt{-1}$ = 4i where 4 is the real number and i can represent $\sqrt{-1}$

Therefore $\sqrt{-16}$ = 4i

• $\sqrt{-25}$ can be written as $\sqrt{25}x\sqrt{-1}$ = 5i where 5 is the real number and i can represent $\sqrt{-1}$

Therefore $\sqrt{-25}$ = 5i

• $\sqrt{-5}$ can be written as $\sqrt{5}x\sqrt{-1}$ where $\sqrt{5}$ is a real number and i can represent $\sqrt{-1}$

Therefore $\sqrt{-5}$ = i$\sqrt{5}$

The Complex Numbers are written in the form of a + bi where a and b are real numbers and i is an imaginary number.

For Example

• 3 + 4i Where a is 3 and b is 4 and i is the imaginary number $\sqrt{-1}$.

• 2i Can be rewritten as 0 + 2i where a is 0 and b is 2 and i is the imaginary number $\sqrt{-1}$.

The key to the Complex Numbers is the understanding that i is an imaginary number that is defined as $\sqrt{-1}$ and that the accepted form is a + bi where a and b are real numbers and i is the imaginary component.

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