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MAT 85 – Introduction to Mathematics for

Computer Science

Lecture #1 – Operator Properties & Intro to Complex Numbers



MAT85 – Introduction to Mathematics for Computer Science Lecture 1

© 2014, DigiPen Institute of Technology. All Rights Reserved. 2

Operator PropertiesThis lecture is a pretty simple one and aims at refreshing some properties that could

always come useful when manipulating mathematical formulas in general, and more specifically

throughout this course and the next.

The lecture deals with the four elementary arithmetic operators – addition, subtraction,

multiplication and division - and their properties. It will be divided as follows: the operator

discussed will be presented and then its properties will be listed, with the property defined and

proven.

Operators are basically functions that represent a certain mathematical operation. They

are applied on what is called operands, which are the quantities on which the operator is applied.

With this basis in place, let’s start talking about each of the operators individually, starting with

the first, addition.

Addition:We start with the first operator, which is the addition operator, which is denoted by the + “plus”

sign. Adding any two numbers x & y is mathematically represented as:

+

At its simplest interpretation, addition can be viewed as a way in which sets of objects are

combined together to form a larger set of objects, whether that set is height, length, width or any

other quantity really.

Addition has several properties which are the following:

Commutative

Associative

Distributive (over multiplication)

Identity element

Inverse element

Let’s go over each one of these properties and discuss how it can be useful to us next.

Commutative:

What this property basically says is that you can interchange the order of the operands, and the

result of the operation (in this case, addition), will remain the same. In the case of addition, the

right terminology would be that changing the position of the addends would not change the

answer of the sum.

In general notation, commutative means that





+ = +

Example:

This is easily verified by choosing x & y as any two values, say 2 & 3 and plugging the numbers

in:

2 + 3 = 3 + 2 = 5.

Associative:

The associative rule states that if an operator is applied on more than two operands, the order

with which the operator is applied does not affect the outcome of the computation.

Mathematically speaking, this is expressed below as:

+ + = + ( + )

Example:

This can be clearly seen in the case of addition by simply adding any three numbers such as 3, 4,

& 5.

3 + 4 + 5 = 7 + 5 = 12

3 + (4 + 5) = 3 + 9 = 12

Naturally, this property can be extended to more than 3 numbers as well.

Distributive (over multiplication):

Addition is distributive when multiplication is involved. The correct term is multiplication is

distributive over addition (and subtraction). What this means, is that if we have two numbers that

are added and their sum is multiplied by a third number, this quantity can be calculated in

another way by multiplying the third number with each of the first two numbers, and then adding

the resultant of those two multiplications. This is expressed mathematically as:

+

∗ =

∗ +

∗

Example:

4 + 7 ∗ 3 = 11 ∗ 3 = 33

4 ∗ 3 + 7 ∗ 3 = 1 2 + 2 1 = 3 3





The property holds!

It’s important to note, that without parenthesis that multiplication and division will happen

before addition and subtraction, however in the first example because 4 + 7 is in parenthesis itwill be added first; this is due to the order of operations. The order of operations (or operator

precedence) is a rule used to clarify which operands should be performed first in a given

mathematical expression. This will be discussed in much more detail in your computer science

classes in the future as it a fundamental thing to know in programming.

Identity element:

The identity element is an element that when used on a number with an operator, will not cause

that number to either increase or decrease, that is, it will remain the same. For the case of

addition, it is very clear to see that the identity element is none other than “0”:

+ 0 = 0 + =

Inverse element:

Unlike the identity element which doesn’t change the number’s value, the inverse element

negates the element being operated on. For addition, the inverse element is simply the negative

of the number used:

+ − = − + = 0

Example:

= 8 → 8 + −8 = −8+ 8 = 0

The→ symbol stands for implies or if .. then, so in the above example we are saying if x is equal

to 8, then the following is true. This could also be written without the = part of the equation.





SubtractionSubtraction is the second operator to be discussed. It is seen as having an original quantity and

taking away a certain quantity from it, the opposite of addition. The – (minus) sign represents it:

−

Subtraction has the following properties:

Distributive (over multiplication)

Inverse element

Let’s go over each one of these properties and discuss how it can be useful to us.

Distributive (over multiplication):

Just like with addition, multiplication is distributive over subtraction. This is expressed as:

− ∗ = ∗ − ∗

Example:

6− 2 ∗ 5 = 4 ∗ 5 = 20

6 ∗ 5 − 2 ∗ 5 = 30 − 10 = 20

Inverse element:

In the case of subtraction, the inverse element is the number itself, because when we have a

number and we subtract it from itself we will get zero:

− = − = 0

Example:

5 − 5 = 5 − 5 = 0

Unlike addition, subtraction is neither commutative nor associative. To prove this, let us consider

an example.

4 − 2 = 2

2− 4 = −2





2 and -2 are not equal (2 ≠ −2) hence the commutative property doesn’t hold. Note that in

mathematics, if we want to disprove a statement, it is enough to give one counterexample and

this would be enough to prove the statement false.

9− 2 − 4 = 7 − 4 = 3

9− 2− 4 = 9 − −2 = 9 + 2 = 11

Once again, once simple counterexample is enough to prove that subtraction is not associative

either.

Also, unlike addition, subtraction has no identity element:

− 0 =

0 − = −

Obviously, the two terms are not equal; hence zero is not an identity element for subtraction. No

other element exists such that:

− = − =





MultiplicationMultiplication is an operation that involves scaling a number by another number. It is

represented by the “*” sign (asterisks), “∙” sign (dot), or the “” (times) sign. That means the

following three statements mean the same thing:

∗

∙

What it does is simply add X to itself Y times. So 5 6 is equivalent to saying

5 + 5 + 5 + 5 + 5 + 5 = 30

Multiplication has the following properties:

Commutative

Associative

Distributive

Identity

Zero element

Inverse element

Let’s go over each of these properties and discuss how we can use them.

Commutative:Just like addition, multiplication is commutative:

∗ = ∗

Example:

4 ∗ 3 = 12 = 3 ∗ 4

Associative:

The associative property applies to the multiplication operator:

∗ ∗ = ∗ ( ∗ )

Example:

3 ∗ 2 ∗ 5 = 6 ∗ 5 = 3 0

3 ∗ (2 ∗ 5 ) = 3 ∗ 10 = 30





Distributive:

As we saw in the addition and subtraction section, multiplication is distributive over both of

these operators:

+

∙ =

∙ +

∙

− ∙ = ∙ − ∙

Identity:

The element that does not change the value of a number when multiplied by it is 1:

∗ 1 = 1 ∗ =

Zero element:A special property for multiplication called the zero element states that any number multiplied by

zero will give us a result of zero, cancelling out what it was given:

∗ 0 = 0 ∗ = 0

Inverse element:

The inverse element of a number is called its reciprocal. To get the reciprocal of a number, you

just need to divide 1 by the number. If you multiply any number by its reciprocal, you will get 1as the result:

1∙

1

=

1

∙1

=

= 1

Simplified, we can write the following:

∙1

=

1

∙ = 1





Examples:

7 ∙1

7= 1

1

7

∙ 7 = 1





DivisionJust as subtraction is the opposite of addition, division is essentially the opposite of

multiplication. It is represented by the / or ÷ signs.

Division is neither commutative nor associative, but it does contain the following property:

Inverse element

Let’s see how we can make sure of that now.

Inverse element:

In division, the inverse element of any number is the number itself (zero not counted):

=

= 1

Division is neither commutative nor associative, and has no identity element. This can be shown

with one counterexample.

6

3= 2;

3

6= 0.5 → ℎ

5

1= 5,

1

5≠ 5, 1





Summary

Property/Operator Addition Subtraction Multiplication Division

Commutative + = + ------ . = . ------

Associative + + = + ( + )

------ .. = . (. )

------

Distributive + . = . + .

− . = . − .

+ . = . + .

------

Identity + 0 = 0 + =

------ . 1 = 1. = ------

Zero element ------ ------

. 0 = 0 . = 0------

Inverse element + −= −+ = 0

− = − () = 0

.1

=

1

. = 1

=

= 1





Introduction to Complex Numbers

Is it possible to have the square root of a negative number? The answer is yes. And this is where

the concept of complex numbers comes into play! One can take the square root of a negative

number, but it involves using a new number denoted as “” and it means imaginary, incomparison to real numbers. This imaginary number is defined as:

= −1

ℎ, 2 = −12

= −1

A few examples will help illustrate how one can manipulate a negative under the square root.

Simplify the examples below:

a-

−9 = −1 ∗ 9 = 9 ∗ −1 = 3 ∗

b- −27 = 9 ∗ 3 ∗ −1 = 3 3 ∗

Just like with real numbers, imaginary numbers can be added together:

a- Addition: 4 + 8 = (4 + 8) = 12

b- Subtraction: 12 – 5 = (12 – 5) = 7

c-

Multiplication: 2(5) = (2.5)(. ) = (10)(2

) = −10

A good way of manipulating the complex number is noticing the following:

2 = −1

3 = 2 = −

4 = 22 = −1−1 = 1

This can go on for any power of .

Example:

7 = 2. 2. 2. = −1−1−1 = −

This can be done in a shorter way as follows:





64004 = 64000+4

= 416000 . 4

= 11

= 1

This is because 4 ∗ = 1 4 = 1

Complex numbers definition

The imaginary number is indeed very useful since it helps us define what a complex number is.

By definition, complex numbers have two parts, a real part and an imaginary part denoted as:

+

Where and are both real numbers, but since has an attached to it, it forms the imaginary

part of the term + . Just like with real numbers, complex numbers can be added, subtracted,

multiplied and divided.

Examples:

Simplify the following:

a- 2 − 4 − 5 − 8

2 − 4 − 5 − 8 = 2 − 4 − 5 + 8

= 2− 5 + 8 − 4

= −3 + 4

b- 2 – 45 − 8 2 – 45 − 8 = 25 − 28 − 45 − 4−8

= 10 − 16 − 20 + 322

= 10 − 36 + 32−1

= −22− 36





c- 9

4=

2.25

To get rid of the in the denominator, all we have to do is multiply the numerator and

denominator by an :

2.25

=

2.25

2

=2.25−1

= −2.25





Complex Conjugates

A very useful property with complex numbers is that when you multiply a complex number by

its complex conjugate you get rid of the imaginary part. Before we see how this happens, first

let us define what a complex conjugate is.

The complex conjugate of a complex number + is

−

So all we have to do to get the complex conjugate is to invert the sign of the imaginary part of

the complex number. Now, if we multiply a complex number with its complex conjugate, this is

what happens:

+ − = − + −

= 2 − + −22

= 2 + 2

And just like that, we were able to get rid of the imaginary part. Let us see that in example:

Examples:

a- Multiply 3− 2 by its complex conjugate and simplify:

3− 23 + 2 = 32 + 22

= 13





b- Get rid of the imaginary part in the denominator of the following:

a. 1+

3−2

1 + 3 − 2

=1 +

3 − 2∗

3 + 23 + 2

= 1 + 3 + 232 + 22

=3 + 2 + 3 + 22

13

=3 + 5 − 2

13

=1

13+

513

1 Operator Properties Intro to Complex Numbers Revised

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