# T H U t h th

We use a decimal  number system – it has a base of 10.

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, or if the number is represented to a given number of decimal places.

Every time a digit moves to the left, it increases in size by  a factor of 10.

102 = One hundred, no tens, two units.
This is read one hundred and two point .

3300 = three thousands, three hundreds, no tens, no units.
This is read three thousand, three hundred.
Zeroes are put in as place holders.

Every time a digit moves to the right, it idecreases in size by  a factor of 10.

102.35 = One hundred, no tens, two units, 3 tenths, 5 hundredths.
This is read one hundred and two point three five.

3020.020 = three thousands, no hundreds, two tens, no units, no tenths and two hundreths.
This is read three thousand and twenty point zero two.

# x 10   x100  x1000

• Multiply by 10   -  Move all the digits one place to the left.

Examples

102.35 x 10 = 1023.5           10352.3577x 10 =103523.577      30 x 10 = 300   50230 x 10 = 502300

• Multiply by 100 -   Move all the digits two places to the left.

e.g    102.35 x 100 = 10235          10352.3577x 100 =1035235.77

30 x 100 = 3000         50230 x 100 = 5023000

• Multiply by 1000 -   Move all the digits three places to the left.

e.g    102.35 x 1000 = 102350          10352.3577x 1000 =10352357.7      30 x 1000 = 30000      50230 x 1000 = 50230000

# ÷  10   ÷ 100  ÷ 1000

• Divide by 10 -   Move all the digits one place to the right.

e.g  102.35 ÷ 10 = 10.235          10352.3577 ÷ 10  =1035.23577

30 ÷ 10  = 3                        50230÷ 10  = 5023

• Divide by 100 -   Move all the digits two places to the right.

e.g    102.35 ÷ 100 = 1.0235          10352.3577 ÷ 100  =103.523577

30 ÷ 100  = 0.3                        50230÷ 100  = 502.3

• Divide by 1000 -   Move all the digits three places to the right.

e.g    102.35 ÷ 1000 =0.10235          10352.3577 ÷ 1000  =10.3523577

30 ÷ 1000  = 0.03                        50230÷ 1000  = 50.23

When we add numbers, we are adding up the totals in each of the units, tens, hundreds etc columns.
Since base 10 only allows the single digits 0 to 9, any extras are carried forward to the next column.

Examples

113 +65 =178  224 +102=326  7+6=13 27+6=313=33

To add numbers together, it is easier to write them out underneath each other.
Start from the units column.  Make sure that the numbers stay in line !

# - Subtraction

To subtract two numbers,

• Line them up, placing the first number on top and the second number underneath.
• Keep the digits in line, matched from the units.
• Then do top number subtract bottom number.

You should then find the top number.

Examples

Calculate a) 178 – 65             b) 326 -102

check 113 +65 =178    224 +102=326

## Order is important.

123-89 is not the same as 89-123 !!!

Problems often occur when the bottom digits are bigger than the top digits.
The temptation then is to switch to bottom take top.
This is wrong.

Example

The solution to the sum "Calculate 123 -89"

is often given as

This is clearly incorrect, since adding the  answer to the bottom number gives
166 +89 =255

There are many methods available to find the correct solution.

### Decomposition

check 34 +89 =123

### Equivalence

As long as the same thing is done to both numbers, the original sum can be changed into an easier one.

123 – 89   is the same as 124 – 90, since 1 has been added to both numbers

124 -90 = 34 so 123 – 89 =34

### Borrow and payback

This is based on a combination of equivalence
and decomposition.

# X Multiplication

Twenty times table

18 x 17 = 306

5 x 30 = 5 x (3 x 10) = (5 x 3) x 10 = 15 x10 = 150

• To multiply large numbers, write them down underneath each other.

Then

• Times the top line by the bottom right hand number.
• Take a new row. Start with a zero for each place moved ( HTU etc)
• Times the top line by the next bottom number.
• Keep going until all bottom numbers have been used.

Examples

### Decimal Multiplication

To multiply decimals,  write them down underneath each other.

Then
*   Ignore the decimal point  .

• Carry out sum as above.
• Count the number of decimal digits.

Starting  at right hand end of  answer, count that number of digits to the left.
Place decimal point

Example

## Egyptian multiplication

With this method, you only need to be able to double numbers and add.

Example
Calculate 17 x 27

First, set up a table

To complete the table,  double each entry

This table will now allow us to find our answer by adding the relevant entries together.

27 = 16 +8 +2 +1
so adding these entries will give

17 x 27 = 272 +136+34 +17 = 459

Example
Calculate 45 x 36

45 x 36 = 1440 +180 =1620

### Grid Multiplication

This method is often used in primary school.
It basically breaks the numbers down into a grid, then the individual components are multiplied and finally added together.

Example 45 x 36

### Gelosia Multiplication

This is a method from the middle ages.
Draw a grid, complete with diagonals.
Multiply the individual components, putting any tens above the diagonal.

Example 45 x 36

Answer 45 x 36 = 1620

4 x 3 = 12, so write 1 above the diagonal and 2 below.

Complete for the rest.
Don’t forget to carry if necessary!

Sometimes this method is written the other way:

# ÷ Division

### Short Division

Tip : First write out the ten times table  for the number you are dividing by, or use the Egyptian multiplication method shown above.

### Dividing  a decimal

Divide as normal. Instead of writing down a remainder, put a decimal point in the answer ( in line with the question) and put a zero after your remainder.
Then keep on going  until you have no remainders left, or have reached a suitable point.

e.g. For an answer to 3 decimal places, work it out to 4 decimal places first.