The Imaginary Unit i

---

You have learned that some quadratic equations have no real solutions. For instance, the quadratic equation x² + 1 = 0 has no real solution because there is no real number x that can be squared to produce -1. To overcome this deficiency, mathematicians created an expanded system of numbers using the imaginary unit (i), defined as

or

A multiple of the imaginary unit, like 2i, 3i, or -7i, is called an imaginary number.  By adding real numbers to imaginary numbers, a set of complex numbers is obtained.  A complex number can be written in the standard form a + bi.

Example

Find the standard form of the complex number .

 

In standard form a + bi, the real number a is the real component, and the number bi (where b is a real number) is the imaginary component of the complex number.

  

Equality of Complex Numbers

Two complex numbers a + bi and c + di are only equal to each other when their real and imaginary components are both equal.

a + bi = c + di  if and only if  a = c and b = d.

 

Addition and Subtraction of Complex Numbers

You can add or subtract two complex numbers in standard form by adding or subtracting their real and imaginary components separately.

Sum:  (a + bi) + (c + di)  =  (a + c) + (b + d)i

Difference:  (a + bi) − (c + di)  =  (a − c) + (b − d)i

 

Example

 

 

Multiplying Complex Numbers

Multiplying complex numbers can be done using the distributive property.  If you multiply a complex number by a real one, distribute the real number over both components.

4(-2 + 3i) = 4(-2) + 4(3i) = -8 + 12i

If you multiply two complex numbers together, use the FOIL method and simplify.  Remember, .

(4 + 6i)(3 − 5i) = (4)(3) + (4)(-5i) + (6i)(3) + (6i)(-5i)
= 12 − 20i + 18i − 30i² = 12 − 2i − 30(-1) = 42 − 2i

 

Complex Conjugates

Pairs of complex numbers in the form a + bi and a − bi are called complex conjugates.  When you multiply them together, you get a real number.

Example

Multiply the complex number 3 + i by its complex conjugate.

The complex conjugate of 3 + i is 3 − i, so the product is

 

Principal Square Root of a Negative Number

If a is a postive number, the principal square root of the negative number -a is defined as

Examples

 

 

Practice

Find real numbers a and b such that the equation is true.

1.  a + bi = -10 + 6i

2.  a + 4i = 13 + bi

 

3.  (a − 1) + (b + 3)i = 5 + 8i

 

4.  (a + 6) + 2bi = 6 − 5i

 

Write the complex number in standard form.

5.  

 

 

6. 

 

 

7. 

 

 

8.  8

 

 

Perform the addition or subtraction and write the result in standard form.

9.  (5 + i) + (6 − 2i)

 

 

10.  (8 − i) − (4 − 3i)

 

 

11.  (13 − 2i) + (-5 + 6i)

 

 

12.  

 

 

Perform the operation and write the result in standard form.

 

13.  (1 + i)(3 − 2i)

 

 

 

 

14.  -6(5 − 2i)

 

 

 

 

15.  8i(9 + 4i)

 

 

 

Answers

1.  a = -10, b = 6

2.  a = 13, b = 4

3.  a = 6, b = 5

4.  a = 0, b = -⁵/₂

5.  4 + 3i

6. 

7. 

8.  8 + 0i

9.  11 - i

10.  4 + 2i

11.  8 + 4i

12.  4 (or 4 + 0i)

13.  5 + i

14.  -30 + 12i

15.  -32 + 72i