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Complex Numbers

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**Definition of a complex number**

If a and b are real numbers, the number a+bi is a complex number and it is said to be written in standard form. If b=0, the number a+bi=a is a real number. If b≠0, the number a+bi is called an imaginary number. A number of the form bi, where b≠0, is called a pure imaginary number. Definition of a complex number

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**The imaginary unit raised to powers**

i is the imaginary unit and is equal to Note the pattern: The imaginary unit raised to powers

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**Venn diagram of the complex number system**

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**Addition and subtraction of complex numbers**

To add (or subtract) two complex numbers, you add (or subtract) the real and imaginary parts of the numbers separately. (3 – i) + (2 + 3i) = (5 + 2i) The sum of two complex numbers can be a real number. Addition and subtraction of complex numbers

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**Multiplication of complex numbers**

You can use the associative property, commutative property, and distributive property of multiplication along with FOIL to multiply complex numbers. (2 – i)(4 + 3i) = 8 + 6i - 4i -3i² = i Multiplication of complex numbers

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**Dividing complex numbers**

To find the quotient of a +bi and c + di where c and d are not both zero, multiply the numerator and denominator by the conjugate of the denominator to obtain Dividing complex numbers

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**Fractal Geometry and the complex plane**

Most applications involving complex numbers are either theoretical or very technical. Many have their basis in fractal geometry which starts with the complex plane. Fractal Geometry and the complex plane

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A fractal art work

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