More with More with Complex NumbersComplex Numbers

Sec. 2.5bSec. 2.5b

Definition: Complex Conjugate

z a bi The complex conjugate of the complex number

is

z a bi a bi What happens when we multiply a complex number byits conjugate???

a bi a bi 22a abi abi bi 2 2a b This is a positiveThis is a positive

real number!!!real number!!!

Practice Problems

2

3 i3

3

i

i

Write the given complex numbers in standard form.

2 2

6 2

3 1

i

6 2

10 10i 3 1

5 5i

5

2 3

i

i

2 3

2 3

i

i

2

2 2

10 15 2 3

2 3

i i i

7 17

13

i

7 17

13 13i

Complex Solutions of Quadratic Equations

2 4

2

b b acx

a

Remind me of the quadratic formula!!!

What’s thisWhat’s thispart called?part called?

It can be used to tell whether the solutions to a particularquadratic equation are real numbers…

The discriminant!!!The discriminant!!!

Discriminant of a Quadratic Equation

2 0ax bx c For a quadratic equation , where a, b, and c

are real numbers and ,0a 2 4 0b ac • If , there are two distinct real solutions.2 4 0b ac • If , there is one repeated real solution.2 4 0b ac • If , there is a complex conjugate pair

of solutions.

Practice Problems

2 1 0x x Solve

21 1 4 1 1

2 1x

a = b = c = 1 Use the quadratic formula!

1 3

2 2i

1 3

2

A complexA complexconjugate pairconjugate pair

Guided Practice

Write the given complex number in standard form.

31 i 21 2 1i i i 2 1i i 22 2i i 2 2i

Guided Practice

Write the given expression in standard form.

2

3

i

i

3

3

i

i

26 3

9

i i

1 2

3 3i

Guided Practice

Write the given expression in standard form.

2 1 2

5 2

i i

i

5 2

5 2

i

i

2

2 2

2 4 2 5 2

5 4

i i i i

i

4 3 5 2

29

i i

220 8 15 6

29

i i i

26 7

29 29i

The Complex PlaneThe Complex Plane

Imaginary Axis

Real Axis

a bibi

a

Imaginary Axis

Real Axis

2 3i3i

2

The Complex PlaneThe Complex PlanePlot u = 1 + 3i, v = 2 – i, and u + v in the complex plane.

Imaginary Axis

Real Axis

1 3u i

2v i

3 2u v i Notice that the twoNotice that the twocomplex numbers, theircomplex numbers, theirsum, and the origin formsum, and the origin forma quadrilateral (what type?)a quadrilateral (what type?)

A Parallelogram!!!A Parallelogram!!!

Definition: Absolute Value of a Complex NumberDefinition: Absolute Value of a Complex Number

The absolute value, or modulus, of the complex number

2 2z a bi a b z a bi , where a and b are real numbers, is

Imaginary Axis

Real Axis

bi

a

z a bi

z

A Few More New FormulasA Few More New Formulas

The distance between the points u and v in thecomplex plane:

d u v

The midpoint of the line segment connecting u and vin the complex plane:

2

u v

A Few More New FormulasA Few More New FormulasFind the distance between u = –4 + i and v = 2 + 5i in thecomplex plane, and find the midpoint of the segmentconnecting u and v.

Distance:

u v 4 2 5i i 6 4i

2 26 4 2 13 7.211

2

u vMidpoint:

2 6

2

i 1 3i

Can we verify theseCan we verify theseanswers graphically?answers graphically?

Whiteboard Problems…

Write the given complex number in standard form.3

3 1

2 2i

331

32

i

213 2 3 3

8i i i 1

1 3 34

i i

213 3 3

4i i i 1

44

i i