Complex Numbers. a + bi Is a result of adding together or combining a real number and an imaginary...

Complex Numbers

a + bi• Is a result of adding together or

combining a real number and an imaginary number

• a is the real part of the complex number.• bi is the imaginary part of a complex

number.• both a and b are real coefficients and i

is equal to the • For example: 6 + 8i

1

Complex Numbers and their graphsGRAPH the following:

-5+4i4+5i

2i-35-i

-2-4i

Complex numbers can be represented as an ordered pair. For example, (5, 6) represents the complex number 5+6i. This ordered pair can be graphed on the complex plane called an "Argand diagram."

The real part of the complex number represents the horizontal coordinate. The coefficient of the imaginary part of the complex number represents the vertical coordinate. The origin corresponds to th point (0, 0) or 0+0i

Modulus

What is the absolute value of -3?│-3│ or 3

Absolute value means the distance from 0 and -3 on the real number line.

Well, there is an absolute value for a complex number as well, and we can find it using our new tool: the argand diagram!

Modulus

• By representing a complex number as a position vector drawn from the origin (0, 0) to the complex number coordinate (a, b), the absolute value of the complex number can be derived.

• This absolute value or MODULUS is equivalent to the vector line's length.

• Develop a formula for the absolute value of a complex number │a + bi│, in terms of a and b

• A sports complex is a building that accommodates two or more sports. A complex number is any number having the a +bi. Explain in your own words why you think these numbers are called complex.

How would you graph i^15

R

Im

Well, first let's look at the cyclical number system of imaginary numbers

R

Im

What is i^0?

1

1 is a real number. The complex number for this value (a + bi) is 1 + 0i

The ordered pair is (1, 0) on the complex plane

Well, first let's look at the cyclical number system of imaginary numbers

R

Im

What is i^1?

i

i is an imaginary number. The complex number for this value (a + bi) is 0 + 1i

The ordered pair is (0, 1) on the complex plane

Well, first let's look at the cyclical number system of imaginary numbers

R

Im

What is i^2?

-1

-1 is a real number. The complex number for this value (a + bi) is -1 + 0i

The ordered pair is (-1, 0) on the complex plane

Are we

starting to see

a pattern

here?

Well, first let's look at the cyclical number system of imaginary numbers

R

Im

What is i^3?

-i

-i is an imaginary number. The complex number for this value (a + bi) is 0 - 1i

The ordered pair is (0, -1) on the complex plane

Well, first let's look at the cyclical number system of imaginary numbers

R

Im

What is i^4?

1

1 is a real number. The complex number for this value (a + bi) is 1 + 0i

The ordered pair is (1, 0) on the complex plane

And now we

are back where we

started...

How would you graph i^15

R

ImWhat is i^15?

(1)(1)(1)(i^3)

i^3 is the imaginary number -i. The complex number for this value (a + bi) is 0 - 1i

The ordered pair is (0, -1) on the complex plane

How would you graph i^-7

R

Im

Well, first let's look at the cyclical number system of imaginary numbers

R

Im

What is i^-1?

-i

-i is an imaginary number. The complex number for this value (a + bi) is 0 - 1i

The ordered pair is (0, -1) on the complex plane

Well, first let's look at the cyclical number system of imaginary numbers

R

Im

What is i^-2?

-1

-1 is a real number. The complex number for this value (a + bi) is -1 + 0i

The ordered pair is (-1, 0) on the complex plane

Well, first let's look at the cyclical number system of imaginary numbers

R

Im

What is i^-3?

i

i is an imaginary number. The complex number for this value (a + bi) is 0 + 1i

The ordered pair is (0, 1) on the complex plane

Are we

starting to see

a pattern

here?

How would you graph i^-7

R

ImWhat is i^-7?

(1)i^-3

i^-3 is an imaginary number i. The complex number for this value (a + bi) is 0 + 1i

The ordered pair is (0, 1) on the complex plane

Rules for graphingNEGATIVE EXPONENTS POSITIVE EXPONENTS

For each successive negative power of i, the

point rotates clockwise by 90 degrees

For each succesive positive power of i, the point

rotates counter-clockwise by 90 degrees

i^0 = 1 i^0 = 1i^-1 = -i i^1 = ii^-2 = -1 i^2 = -1i^-3 = i i^3 = -ii^-4 = 1 i^4 = 1