Complex Number

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

Complex Numbers (Operations)

Operations with Complex Numbers

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.

Adding/Subtracting Complex Numbers

• Combine real terms

• Combine imaginary components

• Distribute negative first (when subtracting)

Adding Complex Numbers

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

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

=10-i

Subtracting Complex Numbers

• (4-5i)-(1-8i)

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

• =3+3i (Combine like terms)

Adding Complex Numbers

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

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

=10-i

​(4+7i)+(8−2)

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

​(3−2i)−(4−2i)

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

(5+8i)+(6−10i)

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

​(−9−5i)−(2−7i)

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

Multiplying Complex Numbers

• Remember to FOIL

• i²=-1

• Simplify as much as possible.

Multiplying Complex Numbers

(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

​Solution Steps

(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

​Solution Steps

(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

​Solution Steps

(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