Powers of i and Imaginary Numbers

A complex number has the form a +bi, where a is the real component and b is the imaginary component.

Ex.

6-3i

0+5i

8+0i

-3+7i

Operations with complex numbers are similar to expressions with variables. You can combine real components with each other and imaginary components with each other.

When multiplying imaginary components, you add exponents, similar to multiplying variables.

• Combine real terms

• Combine imaginary components

• Distribute negative first (when subtracting)

(2+3i)+(8-4i)

=10+-i (Add 2+8 and 3i+-4i)

=10-i

• (4-5i)-(1-8i)

• =(4-5i)+(-1+8i) (Distribute negative)

• =3+3i (Combine like terms)

(2+3i)+(8-4i)

=10+-i (Add 2+8 and 3i+-4i)

=10-i

i)Combine the real and imaginary parts in numbers 4+7i and 8−2i.

4+8+(7−2)i

Add 4 to 8. Add 7 to −2.

12+5i

Subtract 4−2i from 3−2i by subtracting corresponding real and imaginary parts.

3−4+(−2−(−2))i

Subtract 4 from 3. Subtract −2 from −2.

−1

Combine the real and imaginary parts in numbers 5+8i and 6−10i.

5+6+(8−10)i

Add 5 to 6. Add 8 to −10.

11−2i

Subtract 2−7i from −9−5i by subtracting corresponding real and imaginary parts.

−9−2+(−5−(−7))i

Subtract 2 from −9. Subtract −7 from −5.

−11+2i

• Remember to FOIL

• i²=-1

• Simplify as much as possible.

(8+2i)(7-3i)

=56-24i+14i-6i²

=56-10i-6(-1)

=56-10i+6

=62-10i

Solution Steps

(2−i)(3+i)

Multiply complex numbers 2−i and 3+i like you multiply binomials.

2×3+2i−i×3−i2

By definition, i2 is −1.

2×3+2i−i×3−(−1)

Do the multiplications.

6+2i−3i+1

Combine the real and imaginary parts.

6+1+(2−3)i

Do the additions.

7−i

(4+i)(5+i)

Multiply complex numbers 4+i and 5+i like you multiply binomials.

4×5+4i+5i+i2

By definition, i2 is −1.

4×5+4i+5i−1

Do the multiplications.

20+4i+5i−1

Combine the real and imaginary parts.

20−1+(4+5)i

Do the additions.

19+9i

(3−4i)(2+i)

Multiply complex numbers 3−4i and 2+i like you multiply binomials.

3×2+3i−4i×2−4i2

By definition, i2 is −1.

3×2+3i−4i×2−4(−1)

Do the multiplications.

6+3i−8i+4

Combine the real and imaginary parts.

6+4+(3−8)i

Do the additions.

10−5i

(5−3i)(2−5i)

Multiply complex numbers 5−3i and 2−5i like you multiply binomials.

5×2+5×(−5i)−3i×2−3(−5)i2

By definition, i2 is −1.

5×2+5×(−5i)−3i×2−3(−5)(−1)

Do the multiplications.

10−25i−6i−15

Combine the real and imaginary parts.

10−15+(−25−6)i

Do the additions.

−5−31i