Base 10 Arithmetic
Place Value
T H U . t h th
The decimal number system has a base of 10.
Every time a digit moves to the left, it increases in size by a factor of 10.
102 = one hundred, no tens, two units.
3300 = three thousands, three hundreds, no tens, no units.
1023020 = one million, twenty‑three thousand and twenty.
Zeroes are used as place holders.
The decimal point is always next to the units part of the number and only needs to be shown if there is a decimal part.
Every time a digit moves to the right, it decreases in size by a factor of 10.
102.35 = one hundred, no tens, two units, 3 tenths, 5 hundredths.
3020.020 = three thousands, no hundreds, two tens, no units, no tenths, two hundredths.
Note
3020.020 = 3020.02 = 3020.0200000
Zeroes at the end after the decimal point do not matter.
Which number is larger: 45.205 or 45.5?
45.5 is larger, since it has a 5 in the tenths column.
×10, ×100, ×1000
Multiply by 10 → move all digits one place left.
102.35 × 10 = 1023.5
10352.3577 × 10 = 103523.577
30 × 10 = 300
50230 × 10 = 502300
Multiply by 100 → move all digits two places left.
102.35 × 100 = 10235
10352.3577 × 100 = 1035235.77
30 × 100 = 3000
50230 × 100 = 5023000
Multiply by 1000 → move all digits three places left.
102.35 × 1000 = 102350
10352.3577 × 1000 = 10352357.7
30 × 1000 = 30000
50230 × 1000 = 50230000
÷10, ÷100, ÷1000
Divide by 10 → move all digits one place right.
102.35 ÷ 10 = 10.235
10352.3577 ÷ 10 = 1035.23577
30 ÷ 10 = 3
50230 ÷ 10 = 5023
Divide by 100 → move all digits two places right.
102.35 ÷ 100 = 1.0235
10352.3577 ÷ 100 = 103.523577
30 ÷ 100 = 0.3
50230 ÷ 100 = 502.3
Divide by 1000 → move all digits three places right.
102.35 ÷ 1000 = 0.10235
10352.3577 ÷ 1000 = 10.3523577
30 ÷ 1000 = 0.03
50230 ÷ 1000 = 50.23
Addition
When we add numbers, we add the totals in each column. Since base 10 only allows digits 0–9, any extras are carried forward.
113 + 65 = 178
224 + 102 = 326
7 + 6 = 13
27 + 6 = 33
Write numbers underneath each other, starting from the units column.
Subtraction
To subtract two numbers:
- Line them up, first number on top.
- Keep digits aligned from the units column.
- Do top number minus bottom number.
To check your answer, add it to the bottom number — you should get the top number.
a) 178 − 65
113 + 65 = 178 ✔
b) 326 − 102
224 + 102 = 326 ✔
Problems occur when bottom digits are bigger than top digits. The temptation is to switch to bottom minus top — this is wrong.
The solution to “123 − 89” is often incorrectly given as:
This is incorrect because:
166 + 89 = 255
There are many correct methods.
Decomposition
Check: 34 + 89 = 123 
Subtraction by Addition
Equivalence
As long as the same thing is done to both numbers, the sum can be changed into an easier one.
123 − 89 = 124 − 90
124 − 90 = 34
Borrow and Payback
This method combines equivalence and decomposition.
× Multiplication
Twenty Times Table
18 × 17 = 30 65 × 30 = 150
Decimal Multiplication
Ignore the decimal point, multiply normally, then place the decimal point afterwards.
Egyptian Multiplication
This method only requires doubling and adding.
17 × 27
27 = 16 + 8 + 2 + 1
17 × 27 = 272 + 136 + 34 + 17 = 459
Calculate 45 × 36
45 × 36 = 1440 + 180 = 1620
Grid Multiplication
This method breaks the numbers into a grid. Multiply each component and add the results.
45 × 36
Gelosia Multiplication
This is a method from the Middle Ages.
Draw a grid with diagonals. Multiply each component, placing tens above the diagonal. Add along diagonals to read off the answer.
45 × 36
Sometimes written the other way:
÷ Division
Short Division
Tip: First write out the ten times table for the divisor, or use the Egyptian multiplication method shown above.
Dividing a Decimal
Divide as normal. Instead of writing a remainder, place a decimal point in the answer (in line with the question) and continue by adding zeros.
For an answer to 3 decimal places, work to 4 dp first.
Dividing by a Decimal
Long Division
