Section 5.4Imaginary and Complex Numbers

Imaginary Numbers

The result of a square root of a negative number.

To overcome this problem, the IMAGINARY UNIT “i” was created.

Complex NumbersContains a real number and an

imaginary number

Always written in the form a + bi a is the real part, b is the imaginary part

Examples:7 + 2i2.5 – 3i-23 + 6i¼ + 2i5 + 10i

Adding and Subtracting Complex Numbers

Add the real parts, then add the imaginary parts

Examples:

(4+7i) + (-8 + 2i)

(7-5i) + (12-4i)

(12+6i) – (15-3i)

Practice

Page 277, #18-28 even and #38-46 even

Multiplying Complex Numbers

Use the distributive property or FOIL, just as we did with real numbers (remember that i2 = -1)

i(7+i)

2i(10-3i)

(4+i)(3+2i)

(2+3i)(2-3i)

Practice with Multiplication

Page 278, #47-55 ALL

Division with Complex Numbers

Just like with square roots, we do not want to have a complex number in the denominator of a fraction, or division problem.

Example: 53+4i

To simplify this, we use the Complex Conjugate of the denominator.

The Complex Conjugate

of any complex number (a+bi) is (a-bi).

Examples: Complex Conjugate7+4i 7-4i12-3i 12+3i-9-2i -9+2i

5 3+4i

Multiply by the complex conjugate of the denominator over itself

8 2+4i

5+3i1-2i

-5-4i6+7i

Practice

• Page 278, 56-63