OCR GCSE Mathematics

Place value and the four rules

An integer is a whole number, positive, negative or zero. Each digit is worth ten times the digit to its right, so in $52\,318$ the 2 stands for 2 thousands and the 3 for 3 hundreds. Place value is what makes columns work in written addition and subtraction, and it is why multiplying by 10 moves every digit one place to the left.

Adding and subtracting whole numbers by column means lining up digits of the same value and carrying or exchanging when a column runs over or runs short. Both are also worth doing mentally by adjusting to an easier number and correcting afterwards:

Here 98 was rounded up to 100, which takes away 2 too much, so 2 is added back.

Long multiplication

Multiply by each digit of the second number in turn, taking care with place value, then add the results. For $468 \times 37$ the 3 is really 30:

Long division

Long division finds how many times the divisor goes into each successive part of the number. To work out $3\,588 \div 12$, 12 goes into 35 twice (24) leaving 11; bring down the 8 to make 118, and 12 goes into 118 nine times (108) leaving 10; bring down the 8 to make 108, and 12 goes into 108 nine times exactly. So $3\,588 \div 12 = 299$. When a remainder is left over it can be written as a remainder, as a fraction of the divisor, or carried into decimal places by adding zeros after the decimal point.

Negative numbers

On the number line a number further to the left is smaller, so $-9 < -4 < 0$. Adding a negative number moves left and subtracting a negative number moves right, because taking away a debt leaves you better off:

In a product or a quotient, two numbers of the same sign give a positive answer and two of different signs give a negative one.

The same rules hold for a longer chain: a product with an even number of negative factors is positive and with an odd number is negative.

Priority of operations

Everyone gets the same answer from a calculation only because there is an agreed order. Brackets come first, then powers, roots and reciprocals, then multiplication and division working from left to right, then addition and subtraction working from left to right. A fraction bar acts as a bracket around the whole numerator and the whole denominator, and a root sign acts as a bracket around everything under it.

Multiplication and division have equal priority, so $32 \div 4$ is done here after $2 \times 16$ because it comes later in the line reading left to right. The same holds for addition and subtraction: $10 - 4 + 3$ is $6 + 3 = 9$, not $10 - 7$.

The reciprocal of a number is 1 divided by it, so the reciprocal of 8 is $\tfrac{1}{8}$ and the reciprocal of $\tfrac{2}{5}$ is $\tfrac{5}{2}$. Zero has no reciprocal because division by zero is undefined. A number multiplied by its reciprocal always gives 1.

Inverse operations

Addition and subtraction undo each other, as do multiplication and division, and squaring and square rooting (also cubing and cube rooting). Knowing this gives a way to check an answer: if $249 + 98 = 347$ then $347 - 98$ must give 249.

Inverses also solve "I am thinking of a number" problems and missing-digit puzzles by working backwards, undoing the steps in reverse order. If a number is multiplied by 4 and then 7 is subtracted to give 29:

The number was 9, and checking forwards gives $9 \times 4 = 36$ and $36 - 7 = 29$. In a missing-digit sum such as $4\square7 + 3\square5 = 812$, the units column needs $7 + 5 = 12$, so the 2 is written and 1 is carried; the tens column then needs the two digits and the carried 1 to total 11 (to leave 1 written and carry 1 into the hundreds, since $4 + 3 + 1 = 8$), so the missing digits add to 10.

Inverses also justify shortcuts for multiplication: halving one number and doubling the other keeps the product the same, so $35 \times 16 = 70 \times 8 = 140 \times 4 = 560$.