Complex Numbers

Complex Numbers

Complex Numbers

Definition: A complex number is any number, z, which has a real part and an imaginary part:

z = a + bi, where a and b are real numbers and i =  – 1.

Examples of complex numbers:

1 + i, – 1 + 16i, 153 – i, 0.000002 + 5301.7i, – i

The last example given is a complex number with real part zero ; this is called a “pure imaginary” number.

• The entire set of real numbers is embedded within the complex numbers: z = a + 0i
• The entire set of imaginary numbers is embedded within the complex numbers: z = 0 + bi
• Any point (x,y) on the Cartesian plane corresponds to a complex number z = x + yi

(the Cartesian plane then is referred to as the “complex plane”)

• A complex number z = a + bi has another complex number associated with it, called its conjugate , z* = a – bi

(z and z* are conjugates of each other: z is the conjugate of z*)

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Operations with Complex Numbers

Addition

Just add real parts to real parts and imaginary parts to imaginary parts:

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

Examples:

(16 + i) + (7 – 11i) = (16 + 7) + (1 – 11)i = 23 – 10i

(–2 – 5i) + (–3 + 7i) = (–2 + (–3)) + (–5 + 7)i = –5 + 2i

Subtraction

Just subtract real parts from real parts and imaginary parts from imaginary parts:

(a + bi) – (c + di) = (a – c) + (b – d)i

Examples:

(16 + i) – (7 – 11i) = (16 – 7) + (1 – (– 11))i

= (16 – 7) + (1 + 11)i = 9 + 12i

(–2 – 5i) – (–3 + 7i) = (–2 – (–3)) + (–5 – 7)i

= (–2 + 3) + (–5 – 7)i = 1 – 12i

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Operations with Complex Numbers 2

Multiplication

For multiplication, we must use the distributive property:

(a + bi)(c + di) = (a + bi)(c) + (a + bi)(di)

= ac + bci + adi + bd(i 2 ) = (ac – bd) + (ad + bc)i

Notice that we used the definition of i: it is the number which squared gives –1.

Examples:

(–2 – 5i)(–3 + 7i) = ((–2)(–3) – (–5)(7)) + ((–2)(7) + (–5)(–3))i

= (6 – (–35)) + (–14 + 15)i = 41 + i

(7 + ½ i)(–2 + i) = ((7)(–2) – (½)(1)) + ((7)(1) + (½)(–2))i

= (–14 – ½) + (7 – 1)i = – 29 + 6i

2

Properties of i

i 0 = 1, i 1 = i, i 2 = –1, i 3 = –i, i 4 = 1, i 5 = i, i 6 = –1, etc.

i –1 = –i, i –2 = –1, i –3 = i, i –4 = 1, i –5 = –i, i –6 = –1, etc.

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Operations with Complex Numbers 3

Division

For division, we must use the properties of complex conjugates (next slide):

a + bi (a + bi)(c – di) (ac + bd) + (– ad + bc)i

First we multiply the fraction (a + bi) / (c + di) by one. But we choose the form of one to

be the complex conjugate of the denominator: (c – di) / (c – di) . Then just multiply the

numerator factors and the denominator factors. This makes the denominator pure real,

taking advantage of the fact that (c + di)(c - di) = c 2 + d 2 . Examples:

16 + i (16 + i)(–3 – 7i) (–48 + 7) + (–112 – 3)i –41 – 115i

3 – 13i (3 – 13i)(– 5i) – 65 – 15i – 5(13 + 3i) –13 – 3i



=

=

c + di (c + di)(c – di) c 2 + d 2



=

=

=

– 3 + 7i (–3 + 7i)(–3 – 7i) 9 + 49 58

=

=

=

=

5i (5i)(– 5i) 25 25 5

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Properties of Conjugates

You have encountered conjugates before! The special factoring A 2 - B 2 = (A + B)(A – B)

from algebra involved conjugate binomials. There are other uses for conjugates in algebra.

COMPLEX CONJUGATES

• Every complex number z = x + yi has a corresponding conjugate z* = x – yi
• Complex conjugates correspond to points on the complex plane which are “mirror images” of each other (as shown)
• The sum of conjugates is the same as the conjugate of the sum: z* + w* = (z + w)*
• The product of conjugates is the same as the conjugate of the product: (z*)(w*) = (zw)*
• The power of a conjugate is the same as the conjugate of the power: (z*) n = (z n )*

.

{

(x, y)

{

.

(x, –y)

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Polar Form of Complex Numbers

POLAR COORDINATES

On the complex plane, the ray of the positive x-axis is called the pole .

Each point (x, y) on the plane corresponds to an angle  measured counterclockwise from the pole, and a distance out from the origin r along the ray determined by the angle.

.

(r,  )

(x, y)

r



the pole

The relations among x, y, r, and  are:

• r =  x 2 + y 2
•  = tan –1 ( )

• x = r cos 
• y = r sin 

• For the complex number z = x + yi , r is called the magnitude of z and written |z| .
• The angle  is called the argument of z, also written  = arg(z) . The argument of a complex number is not unique; if  = arg(z), then   2  m is also an argument of z.
• The relation z = x + yi becomes z = r(cos  + i sin  ) .

Around and around we go.

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CiS Notation

PRODUCTS and QUOTIENTS

We now have a very wonderful result:

z 1 z 2 = r 1 (cos  1 + i sin  1 ) r 2 (cos  2 + i sin  2 )

= r 1 r 2 (cos  1 cos  2 + i 2 sin  1 sin  2 + i (cos  1 sin  2 + sin  1 cos  2 ))

= r 1 r 2 (cos  1 cos  2 – sin  1 sin  2 + i (cos  1 sin  2 + sin  1 cos  2 ))

= r 1 r 2 (cos [  1 +  2 ] + i sin [  1 +  2 ] )

In words, to multiply two complex numbers, simply multiply their magnitudes and add their arguments .

For division, such as z 1 /z 2 = r 1 (cos  1 + i sin  1 )/ r 2 (cos  2 + i sin  2 ), we need to take a look at the denominator:

r 2 (cos  2 + i sin  2 ) r 2 (cos  2 + i sin  2 ) r 2 (cos  2 – i sin  2 )

r 2 (cos 2  2 – i 2 sin 2  2 )

r 2 (cos  2 – i sin  2 )

1

=

cos  2 – i sin  2

r 2 –1 (cos  2 – i sin  2 )

=

=

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CiS Notation 2

Because of the form of the reciprocal of a complex number we have just investigated,

division of complex numbers has a simple formula, just like multiplication:

z 1 /z 2 = r 1 (cos  1 + i sin  1 )/ r 2 (cos  2 + i sin  2 )

= r 1 r 2 –1 (cos [  1 –  2 ] + i sin [  1 –  2 ] )

NOTATION

Because of the algebraic form of the argument, c os  + i s in  , a complex number

z = r(cos  + i sin  ) may be written as z = r(cis  ) .

This shortened form makes the formulas for multiplication and division much nicer:

• Multiplication of complex numbers: z 1 z 2 = r 1 r 2 cis [  1 +  2 ]
• Division of complex numbers: z 1 /z 2 = r 1 r 2 –1 cis [  1 –  2 ]

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Practice Problems

1. Add 3i and –5.

2. Add 3 – 5i and its conjugate.

3. Subtract 3 – 5i from its conjugate.

4. (4x – yi) + (x + 3yi) =

5. (– 11 – 2i) – (– 9 – 4i) =

6. (4 + 3i)(4 – 3i) =

7. (4 + 3i)(3 + 4i) =

8. (6 + 3i)  (1 – i) =

9. 1 1

10. (1 – i)(4 + 3i)

11. Show that  2 i  2

is a square root of i.

+

=

4 + 3i 4 – 3i

=

(2 + i)(1 + 2i)

( )

+

2 2

12. Two complex numbers are equal

if their real parts are equal and their

imaginary parts are equal.

Solve for a and b so that

( a + bi)(2 – 3i) = (2 + 3i) + (a + bi).

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