Complex Numbers

What is a complex number?

• A complex number can be real or imaginary
• rational or irrational if it's real
• have a repeating decimal if it's irrational
• or be an integer or fraction if it's real

Imaginary numbers

• an imaginary number is the square root of a negative.
• i = √-1
• i² = -1
• if r is a positive real number, then √-r = i√r
• if r = 2, then (i√2)² = i²(2) = -2

Practice

• Simplify: √-50
• -50 = 5 x 5 x 2 x i x i
• one pair of 5's and one pair of i's
• put 5 and i outside of radicand
• 5i√2 is the answer

When solving

• when solving for a variable
• and you are taking the square root of both sides,
• you must have the plus and minus sign

Practice:

• Solve: 2x² + 11 = -37
• subtract 11 from both sides, 2x² = -48
• divide both sides by 2, x² = -24
• take the square root of both sides, remember the plus and minus
• x = 2i√6, x = -2i√6

Writing Complex numbers in standard form

• a + bi
• a is the real part
• bi is the imaginary part, it has the i

Practice

• add and subtract, then write in standard form: (6 - 4i) + (6 + 4i)
• add like numbers,  6 + 6 =12, -4i + 4i = 0
• the answer is 12

More Practice

• multiply complex numbers, then write in standard form: (9 - 2i)(-4 + 7i)
• use foil: -4 x 6 = -36, 9 x 7i = 63i, -2i x -4 = 8i, -2i x 7i = -14i²
• i² = -1, -14 x -1 = 14, rewrite problem: -36 + 63i + 8i + 14
• combine like terms: -36 + 14 = -22, 63i + 8i = 71i
• the answer is -22 + 71i

The End