1.06 The four operations

There are four main operations in arithmetic - addition, subtraction, multiplication, and division. They are all kinds of binary operation, where binary means "two" - they each combine two numbers together in different ways to produce a result. We use them to form expressions and equations with numbers:

Addition

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

If you know how all the single digit numbers add together, you can add numbers of any size by using one of these strategies. Practice your single digit addition here:

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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$1 to the result in the next column along if your sum is $10$10 or more - we say that we carry the $1$1.

Worked example

Example 1

Find $271+95$271+95.

Think: When using the vertical algorithm, we will need three columns - one for units, one for tens, and one for hundreds.

Do: Write the numbers into the table like this:

Then add the numbers in the units column together, and write the result at the bottom:

When we add the tens together, we make $16$16. We write the units of the result down the bottom, and carry the $1$1 to the next column:

Then we add the carried $1$1 to the $2$2 in the hundreds column to finish:

We can write the concluding equation $271+95=366$271+95=366.

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

 

Addition by regrouping

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.

Worked example

example 2

Using regrouping, find $289+76$289+76.

Think: We can write $289=200+80+9$289=200+80+9, and $76=70+6$76=70+6. We can add each part together separately.

Do: 

$289+76$289+76	$=$=	$200+80+9+70+6$200+80+9+70+6	(Split number up)
	$=$=	$200+80+70+9+6$200+80+70+9+6	(Regroup)
	$=$=	$200+150+15$200+150+15	(Add)
	$=$=	$200+165$200+165	(Add)
	$=$=	$365$365	(Add)

 

Subtraction

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

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

$13-7=6$13−7=6 is related to $6+7=13$6+7=13 is related to $13-6=7$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$10 to the top number and take $1$1 away from the top number in the next column.

Worked examples

example 3

Evaluate $953-121$953−121.

Think: Our place value table will need three columns: hundreds, tens, and units.

Do: Write the numbers in the first two rows of the place value table:

Then subtract the bottom number from the top number in each column:

We then conclude that $953-121=832$953−121=832.

Example 4

Evaluate $912-467$912−467.

Think: The place value table will be the same size as the previous example, but this time we will need to borrow tens from later columns as we go.

Do: Write the numbers in the first two rows of the place value table:

The number in the bottom row of the units column is larger than the number in the top row. We borrow $10$10 from the next column along, which lets us perform the subtraction:

In the tens column the top number is now $1-1=0$1−1=0, which is less than $6$6. We borrow $10$10 from the hundreds column and perform the subtraction:

We can then finish the subtraction by subtracting the numbers in the hundreds column:

We then conclude that $912-467=445$912−467=445.

 

Multiplication

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

There is a connection between multiplication and area - you need $15$15 small squares to cover a rectangle that is $5$5 units long and $3$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$3 units long and $5$5 units high requires the same number of small squares to cover it:

Explore this applet to see how two single digit numbers multiply together:

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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.

Worked example

example 5

Evaluate $19\times6$19×6.

Think: Sometimes we don't know how many columns we will need. It's best to over-estimate, so we will use a place value table with three columns: one for hundreds, one for tens, and one for units.

Do: Write the numbers in the first two rows:

When we multiply the top units by the bottom units, we get $6\times9=54$6×9=54. We write the $4$4 units in the bottom row and carry the $5$5:

Then multiply the top tens (just the $1$1) by the bottom units, and afterwards add the carried $5$5:

Notice that instead of writing $11$11 in the tens column, we write $1$1 in the hundreds column and $1$1 in the tens column. We can then conclude that $19\times6=114$19×6=114.

 

Larger multiplications

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

Worked example

example 6

Evaluate $234\times62$234×62.

Think: We will need a thousands column, and may need more. We make sure to include that in our place value table. Since the second number has two digits, we will need two rows for multiplication, and a final row for adding our results.

Do: Write the numbers in the first two rows of the table:

Then multiply the top number by the units digit of the bottom number (which is $2$2):

We then write a $0$0 in the next row on the right, and multiply with the tens digit of the second row (which is $6$6). We add the ten-thousands column that we discover that need as we go:

Now we add the two multiplications together for the final result:

We then conclude that $234\times62=14508$234×62=14508.

 

Multiplication with the area model

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.

Worked example

example 7

Find $86\times53$86×53 using the area model.

Think: We are going to imagine a very large rectangle with side lengths $86$86 and $53$53. We then split up the sides of the rectangle according to their place value.

This is not to scale!

We can then find the area of each individual rectangle by multiplying the lengths of each side:

Finally we add these numbers together for the final result:

$86\times53=4000+240+300+18=4558$86×53=4000+240+300+18=4558.

 

Division

The fourth operation is division, which uses the $\div$÷​ symbol, and is essentially the opposite of multiplication. The expression $15\div3$15÷​3 loosely means "$15$15 split into $3$3 groups", which is the same as the number $5$5. Explore this applet to see how to divide numbers up to $144$144:

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A division equation is (almost) always related to a multiplication equation:

$20\div5=4$20÷​5=4 is related to $4\times5=20$4×5=20 is related to $20\div4=5$20÷​4=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.

 

Worked example

example 8

Evaluate $3961\div17$3961÷​17.

Think: The dividend is $3961$3961, so it goes underneath the bar. The divisor is $17$17, so it goes to the left of the bar.

Do: Form the short division tableau with the dividend and divisor:

The divisor, $17$17, is larger than the first number to the left of the curved line, $3$3. This means we carry the $3$3 across to the next number along to form $39$39:

Now $39\div17=2$39÷​17=2 with $5$5 remainder, because $17\times2=34$17×2=34 and $39=34+5$39=34+5. We write the $2$2 above the bar and carry the $5$5 across to the next number to form $56$56:

We calculate that $56\div17=3$56÷​17=3 with $5$5 remainder, so write a $3$3 above the bar and carry the $5$5 across to form $51$51:

We fill in the last number above the bar with a $3$3, because $51\div17=3$51÷​17=3:

Since there is no remainder at this step, we conclude that $3951\div17=233$3951÷​17=233.