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# Arithmetic

## Arithmetic

Arithmetic involves working with numbers. All our numbers are made from the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 or 9. The value of a digit depends on its place in the number, for example, in 5,291 the 9 is worth 90, but in 5,921 the 9 is worth 900. This is called place value.

When you solve problems that contain numbers by counting or by calculating you are using arithmetic.

There are four different arithmetic operations we use when we calculate. These are addition, subtraction, multiplication, and division. Addition and multiplication are quick ways of counting forward. Subtraction and division are quick ways of counting backward.

When you were very young you probably worked out an addition calculation such as 6 + 3 by counting out six blocks, then counting three more, putting them together and counting them all again. You soon learnt to start from six, and then count on three to reach nine. Later you began to learn addition facts and now you know that 6 + 3 = 9. You know that 60 is six tens and that 30 is three tens because you understand place value. You can use your knowledge of addition facts and place value to work out quickly that 60 + 30 = 90 and that 666 + 333 = 999. If you know all the addition facts to 9 + 9 and understand place value you can work out any additional calculation.

### WHAT IS SUBTRACTION?

Subtraction is the opposite, or inverse, of addition. The calculation 9 – 6 can be done by starting with nine objects and taking six away to leave three, or by comparing towers of nine blocks and six blocks to find the difference, or by starting from nine and counting back six to reach three. If you know that 6 + 3 = 9, then you know that 9 – 3 = 6 and that 9 – 6 = 3.

Understanding place value means that you can work out 999 – 666 very quickly. Knowledge of subtraction facts and understanding of place value help you to work out any subtraction calculation.

Addition and subtraction are inverse operations because one reverses the other.

### WHAT IS MULTIPLICATION?

When a problem involves adding the same number several times, the easiest way to work out the result is to multiply because multiplication is repeated addition. For example, 6 + 6 + 6 + 6 is six four times, or six multiplied by four, which is written as 6 x 4. If you know all the multiplication facts to 9 x 9 and understand place value you can work out any multiplication calculation.

### WHAT IS DIVISION?

The division is the inverse of multiplication. Division involves working out how many times a given number can be subtracted from another number. For example, 24 divided by 6, which is written as 24 ÷ 6, can be worked out by counting the number of times that 6 can be subtracted from 24 before zero is reached. A quick way to do this is to use your knowledge of multiplication facts by thinking how many times does 6 go into 24?

Multiplication and division are inverse operations because one reverses the other.

### WHAT IS COMMUTATIVE LAW?

Addition and multiplication calculations can be done in any order. This is called the commutative law. For example, 3 + 27 has the same sum (30) as 27 + 3, and 4 x 3 has the same product (12) as 3 x 4. Subtraction and division cannot be done in any order, so they are not commutative. For example, 17 – 8 does not have the same result as 8 – 17, and 72 ÷ 9 is not the same as 9 ÷ 72.

### HOW CAN WE CALCULATE IN OUR HEADS?

There are many calculations that we can do in our heads by using a variety of methods to make the calculations easier. For example, an easy way to subtract 19 from a number is to subtract 20 then add 1; an easy way to find 16 + 17 is to double 16 and add 1; an easy way to divide a large number by 5 is to divide by 10 and double the answer; an easy way to multiply by 18 is to multiply by 9 then double the answer. If you use what you know about the numbers involved in a calculation you can often find an easy way to work it out mentally.

HOW CAN WE WRITE DOWN OUR CALCULATIONS?

When a calculation is very difficult to work out mentally we often write it down and work it out on paper. Written methods help us to remember the numbers involved in a calculation and make it easier to work out the answer in stages.

The written method shown here is called long multiplication.