1.01 The four operations

The most fundamental of these is addition, which uses the + symbol. The expression 5+3 loosely means "5 with 3 more", which is the same as the number 8. Think about it as moving from left to right along a number line:

Starting at 5 and taking 3 steps right lands on 8.

For larger numbers, we can use a place value table like this:

Write the two numbers you want to add together, one above the other, in the first two rows. Make sure to match their units, tens, hundreds, and so on in the same column. Add the numbers from right to left, adding 1 to the result in the next column along if your sum is 10 or more - we say that we carry the 1.

We can add more than two numbers together in a single place value table. If any column adds up to more than 10, we need to carry the number of tens into the next column - for example, if a column adds to a number between 40 and 49, we need to carry the 4 into the next column.

Another strategy is to use regrouping, where we split each number up into convenient pieces. We can then add the numbers together in a different order, since reversing the order that numbers appear in an addition expression doesn't change the answer.

Examples

The next operation is subtraction, which uses the - symbol, and is essentially the opposite of addition. The expression 5-3 loosely means "5 with 3 less", which is the same as the number 2. Think about it as moving from right to left along a number line:

Starting at 5 and taking 3 steps left lands on 2.

A subtraction equation is always related to an addition equation, like this:

13-7=6 is related to 6+7=13 is related to 13-6=7.

We can perform subtraction using a place value table as well. Write the first number above the second, and work from right to left, subtracting the bottom number from the top number. If the bottom number is larger than the top number in any column, add 10 to the top number and take 1 away from the top number in the next column.

Examples

Multiplication uses the symbol \times, and is related to addition. The expression 5\times3 loosely means "3 groups of 5", which is the same as the number 5+5+5=15. Notice that the number 3 indicates how many groups of 5 there are.

There is a connection between multiplication and area - you need 15 small squares to cover a rectangle that is 5 units long and 3 units high:

Just like addition, reversing the order that numbers appear in a multiplication expression doesn't change the answer - a rectangle that is 3 units long and 5 units high requires the same number of small squares to cover it:

Once you know how all the single digit numbers multiply together, you can use a place value table to multiply large numbers together. Write one number above the other, and multiply the top numbers by the bottom unit, working right to left.

Examples

To multiply even larger numbers, we use the bottom units digit in exactly the same way. Once that is finished, we write a 0 below our result in the units column and multiply with the bottom tens digit, then add our results at the end.

Another way to multiply large numbers is to use the area model. We imagine a large rectangle split up into smaller ones by place value.

Examples

The fourth operation is division, which uses the \div symbol, and is essentially the opposite of multiplication. The expression 15\div3 loosely means "15 split into 3 groups", which is the same as the number 5.

A division equation is (almost) always related to a multiplication equation: 20\div5=4 is related to \\ 4\times5=20 is related to 20\div4=5.

When dividing larger numbers we will use a procedure called short division. We write the number we are dividing by (the divisor) on the left and the number we are dividing (the dividend) on the right, draw a curved line between them, and a bar across to the right. We will be writing our result (called the quotient) above the bar.

Together this forms a division tableau. We then work from left to right, performing smaller divisions and carrying the remainder along.

Examples

Example 5

Evaluate 1616\div 4.

Worked Solution

Create a strategy

Use short division.

Apply the idea

Form the short division tableau with 1616 as the dividend and 4 as the divisor.

We cannot divide 1 by 4, so we carry the 1 across to the next number along to form 16.

4 can fit into 16\,\,4 times since 4\times 4=16. So we put the 4 in the hundreds place.

Again, we cannot divide 1 by 4, so we carry the 1 across to the next number along to form 16. We put a 0 in the tens place.

4 can fit into 16\,\,4 times so we put the 4 in the units place.

Since there is no remainder at this step, we conclude that 1616\div 4=404

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