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Number: Theory and Examples
An understanding of place value, addition, subtraction, multiplication and division with
whole numbers is essential for all work with numbers. Explanations of some of the terms
that have been used in this document can be found in the glossary on our website. Part 2
of this document contains exercises to practise.
Contents
Area of study Page
1. Recognising place values of whole numbers 1 2. Rewriting numbers in expanded / compact form^2 3. Adding and subtracting numbers with two or more places 2 4. Multiplying numbers ending in 0^2 5. Performing long multiplication 3 6. Dividing whole numbers 4 7. Working out remainders 5 8. Determining an average^5 9. Operating with powers 6 10. Working through an equations using order of operations^6 11. More information 7
1. Place value of whole numbers
Each digit in a number has a place value. The following table shows this order:
Hundred thousands Or 100 000 s
Ten thousands Or 10 000 s
Thousands Or 1000 s
Hundreds Or 100 s
Tens Or 10 s
Units Or 1 s
4 3 1 9 5 6 7 0 2
If we have the number 4319 and concentrate on the digit 4
4 is the face value of the first digit 1 000s is the place value of the digit 4 000 is the total value of the digit
If we have the number 56702 and concentrate on the digit 7
7 is the face value of the third digit 100s is the place value of the digit 700 it the total value of the digit
ACADEMIC SKILLS Division of Student Success
2. Expanded / compact form
Examples:
4 319 can be written in expanded form as 4 x 1 000 + 3 x 100 + 1 x 10 + 9 x 1 56 709 can be written in expanded form as 5 x 10 000 + 7 x 100 + 0 x 10 + 9 x 1 2 x 1 000 + 0 x 100 + 6 x 10 + 3 x 1 written in compact form is 2 063
3. Addition and subtraction
Addition
Example:
Notes 5+8+2 = 15, the 5 is entered and the 1 carries over to the next place, 2+9+5+1 = 17, the 7 is entered and the 1 carries over, 0+7+1 = 8, 1
Subtraction
Examples:
a) 253 – 171 21 1 5 3 -
Notes 3 - 1 = 2 5 – 7 = cannot be done, trade 1 place from the next place along to make 5 into 15, now 15 – 7 = 8 The 2 hundreds has been reduced to 1 hundred (it was traded to do the above subtraction), this leaves 1-1 = 0
b) 632 – 89 65 132 12 -
4. Multiplying numbers ending 0
Examples:
Notes 800 x 4 has two zeros at the end, ignore those zeros. So 8 x 4 = 32 Then add two zeros to the end of your answer. So 800 x 4 = 3 200
a) 6 x 20 = 6 x 2 (0) = 120 b) 800 x 4 = 8 x 4 (00) = 3 200 c) 50 x 100 = 5 x 1 (000) = 5 000 d) 600 x 2000 = 6 x 2 (00000) = 1 200 000
6. Dividing whole numbers
Examples:
a) 516 ÷ 4 or 516 4 Long division method:
Notes Divide 4 into 5 = 1 Then multiply 4 x 1 = 4 Then subtract 5 – 1 = to find left over
Pick up the next digit: the number to be divided into becomes 11 4 into 11 = 2, multiply 4 x 2 = 8 The subtract 11 – 8 = 3 to find left over
Pick up the next digit: then number to be divided into becomes 36 4 into 36 = 9, multiply 4 x 9 = 36 Then subtract 36 – 36 = 0 to find left over or remainder.
We have used up all the digits in the number being divided so the division is finished and there is none left after the final subtraction. So 4 divides into 516 exactly 129 times. Or use the short division method:
1 3
The multiply and subtract is done mentally rather than writing it down.
Notes How many 4s in 5: 1 with 1 left over The 1 joins with the 1 after the 5 to make the number to be divided into 11 How many 4s in 11: 2 with 3 left over The 3 joins up with the 6 to make the number to be divided into 36 How many 4s in 36: 9 with none left over – that’s the last digit so the division is finished.
b) 1256 ÷ 8 or 1256 8 Long division method:
Divide, then multiply, then subtract
or^4
7. Remainders
The remainder represents what is left over after dividing a number.
Examples:
What is the remainder when 212 is divided by 5?
The remainder is 2.
Notes 5 divides in 212: 42 times and 2 left over
or 1
r
the remainder is 2.
Notes It can also be written in decimal or fraction term i.e. 42 25 or 42.
8. Average
Examples:
a) Find the average of 2,5 and 8. Average = (2 + 5 + 8) ÷ 3 = 15 ÷ 3 = 5
d) 9 + 15 ÷ (3 + 2) 5 5 9 + 15 ÷ (3 + 2) ↑ = 9 + 15 ÷ 5 ↑ = 9 + 3 = 12
Notes Work out the brackets first: 3 + 2 is 5 Then work out the division: 15 is divided by 5 is 3 Then back to the start and finish off the addition: 9 + 3 is 12
11. For more information
Visit https://www.csu.edu.au/current-students/learning-resources/build-your-skills/academic-skills- help/numeracy for more information
