# UnderstandingNumbers

## Get Started

This topic will cover how to:

• Nominal/Cardinal/Ordinal numbers;
• Positive/Negative numbers;
• lntegers/Whole numbers;
• Rational/lrrational;
• Prime/Composite;
• Square numbers/Triangular numbers

## Nominal Numbers

Nominal - is a word used to describe the name of the number. "Nominal" simply means"name"! In simplest terms it is the number! Knowing both the face and place values helps us read and say the names of numbers.

If numbers were people they could use nominal numbers to have a conversation, perhaps like this...

"Hi. My name is seven. What's yours?" "Hi. Nice to meet you. My name is ninety four."

## Cardinal Numbers

Cardinal numbers = counting numbers.

What do they do? They help us to describe how many.

What are they? Cardinal numbers only include whole numbers, not parts of numbers such as decimals and fractions.

NOTE: Knowing both the face and place values of numbers helps us read, say and count (cardinal) numbers.

EXAMPLES OF CARDINAL NUMBERS

1 = One

2 = Two

3 = Three

423 = Four hundred and twenty three

This of course, can continue forever!

PRACTICAL EXAMPLE OF USING CARDINAL NUMBERS

Eg. I have lost four iPhones this year!

## Ordinal Numbers

Ordinal Numbers = ordered numbers.

What are they? An ordinal number refers to the numerical order, sequence or position of something.

It would be useful for us to understand this word better if it was spelled "Ordernal" - but it isn't!

It does mean exactly the same thing though. Order means sequence and position.

How do ordinal numbers differ from Cardinal?

As you learnt last slide, Cardinal numbers refer to howmany of something. Ordinal refer to the position of something when it is in a seguence.

PRACTICAL EXAMPLES OF ORDINAL NUMBERS

"Robert came first (1st) in the running race."

"I am the second (2nd) son in my family."

"Your ticket at the concert is for the (3rd) third row and the (11th) eleventh seat.

"The seventh (7th) Prime Minister of Australia was Billy Hughes."

## Positive Numbers

Positive Numbers = these are any number greater than zero (0). Therefore any number which is greater or bigger than zero is called a positive number

Positive numbers can include fractions and decimal numbers.

How do positive numbers appear on a number line?

We often use a basic number line to understand the interaction between positive numbers and negative numbers. Below is a number line that will assist you to understand positive numbers in a graphical sense.

Understanding number signs- Positive numbers may have a sign (+) and can be shown as such +4,or+5 etc If there is no symbol then it is automatically a positive number e.g. 5 is the same as +5

## Negative Numbers

Negative Numbers = all numbers that are less than zero (0).

Another way to say this is: a negative number is any number smaller than zero(0).

Negative numbers can also include fractions and decimal numbers.

How do negative numbers appear on a number line?

Similar to the number line you saw in the last slide, below is a number line that will assist you to understand negative numbers in a graphical sense.

Understanding number signs Negative numbers are written with a negative sign before them. For example negative five is written as -5. The dash before the five indicates that it is negative.

## Whole Numbers And Integers

Whole Numbers = A whole number is any complete (whole) number above zero such as 1,2,3,4,5 and so on.

Whole numbers can only be positive, and they cannot be a fraction or decimal...

lntegers = Complete numbers [not fraction or decimal] which can be positive, negative, or zero! This is the difference between whole numbers and integers, an integer is any complete number, whereas a whole number must be above zero.

EXAMPLES OF INTEGERS

-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5

Negative IntegersPositive Integers

## Examples Of Integers

Integers

We see and use positive and negative integers in our environment all the time. Below are some examples.

The weather report

It was minus 5 degrees today in Japan!

A bank account

I have negative \$125 in my bank account!

Sales report

Today we sold 25 more than usual.

## Rational Numbers

A Rational Number is any number that can be written as a fraction and is finite (fixed and not on going). Possibilities include the following:

1) Positive and negative integers. All integers can also be written as a fraction by placing the integer itself over (the number) one.

Examples: Positive integers written as fractions

7 = 7/15 = 5/19 = 9/1

AND

Negative integers written as fractions

-7 = - 7/1-5 = - 5/1-9 = - 9/1

2) Fixed quantity or an exact amount, even if a fraction or decimal.

e.g.

1/3 0.1428571428573/4 0.756/8 0.75

0.142857142857
0.75
0.75

## Irrational Numbers

Irrational Numbers include any number that is not Rational!

Irrational numbers include numbers that are not a fixed quantity/amount and cannot be expressed exactly as a common fraction.

Irrational numbers cannot be represented as ratios or proportions (rational numbers can be) because this would be impossible.

How to identify irrational numbers? lrrational Numbers never end such as: non recurring decimal numbers. For example:

Pi = 3.141592... (the numbers in the decimal places keep going forever, in ever increasing accuracy). Click to see how pie goes on indefinitely.

## Factors & Multiples

Factors are numbers that "go into" (can be divided into) a number exactly. This means they have no part remaining or left over when shared into a number.

Example: The integer 5 divides exactly into the larger integer 15, so we can say that:

5 is a factor of 15.

First Way: one way we can check this is to count up by 5. If we count by "lots of" 5 then we will land on 15, exactly. (3 lots of 5 = 15)

5plus 5 more10plus 5 more15

xxxxxxxxxxxxxxx

Second Way: another way to find factors of a number is to share the larger number by lots of a smaller numbers (divideit). If the answer is a whole number with no left over, remainder or decimal, then you have found another factor!

For example: 15 + 5 = 3 -> you can see here there is no decimal left over. The answer is a whole 3.

Second Way: another way to find factors of a number is to share the larger number by lots of a smaller numbers (divide it). If the answer is a whole number with no left over, remainder or decimal, then you have found another factor!

For example: 15 + 5 = 3 -> you can see here there is no decimal left over. The answer is a whole 3.

## Factors (cont)

Example: what are the factors of 6?

Try this activity.

a) Can 6 be divided by one? Count along line by 1 YES!

b) Can 6 be divided by two? Count along line by 2 YES!

c) Can 6 be divided by three? Count along line by 3 YES!

d) Can 6 be divided by six? Count along line by 6 YES!

The numbers 1, 2, 3, 6 all 'land on' 6 so are all factors of 6.

Question - "Is the integer 4 a factor of 6?" WHY?

NO! Because 6 is not divisible by 4 exactly.

## Factors (cont)

So, the "factors of the number 12 are (1,2,3,4,6,and 12)."

The number one will always be a factor of ALL whole numbers.

The number itself (in this case 12) will always go into itself so will always be a factor.

Therefore any number will always have at least one and the number itself as factors!

## Multiples

A multiple is the number any factor will "go into".

Example:

The factors of 15 are 1,3,5, and 15.

Therefore we can say that: 15 is a multiple of 1, 3, 5, and 15.

Q. Is 28 a multiple of 7?

When we multiply a factor of any number by another factor of the same number and we get a multiple, i.e. factor x factor = multiple.

Example: e.g. 7 and 4 are factors of 28. Yes, 28 is a multiple of 7.

So, factor x factor 7 x 4 = 28 (the multiple)

Q. Is 28 a multiple of 4? Yes, 28 is a multiple of 4.

Q. What are the multiples of 7? The multiples of 7 are 7,14,21,28.35...

## Prime Numbers

Prime Number : A prime number has only two factors, the number one (1) and the number itself....and no more.

Example:

1. The integer 7 is a prime number as it only has the two factors, 1 and 7.

2. The only factors of 5 = 1 and 5, is 5 a prime number?

3. Is 11 a prime number? What factors does 11 have?

7 is therefore a prime number
5 is therefore a prime number

## Composite Numbers

Composite numbers:These numbers are called COMPOSITE numbers = have more than two factors. These are the opposite to prime numbers from the last slide.

Example:

1. The integer 9. Is 9 a composite number? What are its factors?

Factor of 9:

2. The only factors of 5 = 1 and 5, is 5 a prime number?

Factor of 12:

Nine is NOT a prime number because it has factor(s) other than one (1) and itself (9), It also has integer three (3) as a factor.
Therefore 9 is a composite number.

Twelve has more than 1 and 12 as factors. It also has 3,4, and 6 as factors. This makes 12 a composite number.
Therefore 12 is a composite number.

## Square Numbers

Square numbers: If represented as objects e.g. dots on a page, can be placed into a square shape, with equal sides (x4) and equal number of rows and columns.

The integer nine (9) as shown above IS a square number because we can represent it as a square shape.

Here are many square numbers. Can you think of some other square numbers?
Here are some....

4 , 9, 16 , 25

TIP: when we multiply a number by it self, we get a square number.

Lets take a look on the next slide.

4 , 9, 16 , 25

## Square Numbers (cont)

Multiply a number by itself to get a square number.

a) 2 x 2 = 4b) 3 x 3 = 9c) 4 x 4 = 16d) 10 x 10 = 100

## Triangular Numbers

Triangular numbers: These can be represented in the shape of a triangle.

Example:

1. Is three (3) a triangular number?

le. Will three dots fit triangularly here?

2. Is six (6) a triangular number?

le. Will six dots fit triangularly here?

## Triangular Numbers (cont)

Triangular numbers can be explored a little further.

Count the numbers of smiles on each row below and see if you can find a pattern, [tip: be sure to try to figure it out before clicking to reveal].

## End Topic

CONGRATULATIONS ON COMPLETING THE TYPES OF
NUMBERS TOPIC.

This topic has covered the following items:

• Nominal/Cardinal/Ordinal numbers;
• Positive/Negative numbers;
• lntegers/Whole numbers;
• Rational/lrrational;
• Prime/Composite;
• Square numbers/Triangular numbers