CCEA GCSE Mathematics

Place value and decimals

Each digit in a number has a value set by its position. In $4\,036.275$ the 4 is worth 4 thousands, the 0 is a placeholder with no hundreds, the 3 is 3 tens, and after the decimal point the 2 is 2 tenths, the 7 is 7 hundredths and the 5 is 5 thousandths. Moving a digit one place to the left makes it ten times bigger, and one place to the right makes it ten times smaller, which is why multiplying by 10, 100 or 1000 moves every digit that many places left.

To compare decimals, line up the decimal points and compare digit by digit from the left, adding zeros if that helps. Comparing $0.35$, $0.305$ and $0.4$ as $0.350$, $0.305$ and $0.400$ gives the order $0.305 < 0.350 < 0.400$. The length of a decimal says nothing about its size.

Decimal notation for money keeps two decimal places: £3.50, not £3.5, and £0.80 or 80p, never £0.80p.

Adding and subtracting decimals

Write the numbers underneath each other with the decimal points in a line, fill any gaps with zeros, and add or subtract as with whole numbers, putting the point straight down in the answer.

Multiplying and dividing decimals

To multiply, ignore the points, multiply the whole numbers, then put the point back so that the answer has as many decimal places in total as the two numbers being multiplied. For $0.6 \times 0.07$, work out $6 \times 7 = 42$; there are one and two decimal places, so three in the answer, which gives $0.042$.

To divide by a decimal, multiply both numbers by the same power of 10 so that the divisor becomes a whole number. This does not change the quotient, because $\dfrac{a}{b} = \dfrac{10a}{10b}$. So $7.2 \div 0.09 = 720 \div 9 = 80$.

Negative numbers

On a number line, numbers to the right are bigger. So $-7 < -2 < 0 < 3$, and the number with the larger size is the smaller one when both are negative.

For addition and subtraction, think of movement along the line. Subtracting a negative number moves to the right, exactly like adding the positive, so

For multiplication and division, two numbers with the same sign give a positive answer, and two with different signs give a negative one. So $(-4) \times (-6) = 24$, $(-20) \div 5 = -4$ and $(-18) \div (-3) = 6$. This rule follows through for any number of factors: an even number of negative factors gives a positive product.

Inequality symbols

The symbol always points at the smaller quantity. A statement such as $-3 \le n < 2$ describes every number from $-3$ up to, but not including, 2, so the integers that satisfy it are $-3, -2, -1, 0, 1$.

Priority of operations

When a calculation has several operations, the order is fixed by convention:

1. brackets first, and the top and bottom of a fraction count as if each were in brackets;
2. then powers and roots;
3. then multiplication and division, from left to right;
4. then addition and subtraction, from left to right.

Reciprocals fit in the same way. The reciprocal of a number $n$ is $\dfrac{1}{n}$ (written $n^{-1}$), so the reciprocal of 4 is $0.25$ and the reciprocal of $\tfrac{2}{3}$ is $\tfrac{3}{2}$. A number multiplied by its reciprocal gives 1. Zero has no reciprocal.

Using a calculator

A scientific calculator follows the same priority rules as written working, so it is the keystrokes that must match the structure of the calculation. Fractions and roots need brackets round the whole of the top or the whole of the part being rooted. For $\dfrac{18 - 6}{2 + 4}$ the calculator needs $(18-6)\div(2+4)$, because $18-6\div2+4$ would give $19$. The fraction key, the power keys and the square root key can keep the structure without extra brackets, and it is always worth checking the answer against a rough estimate.

Squares, cubes and roots

A square number is a whole number multiplied by itself: $1, 4, 9, 16, 25, 36, \ldots$. A cube number is a whole number multiplied by itself twice: $1, 8, 27, 64, 125, \ldots$. Both come from index notation, $n^2$ and $n^3$, and powers of 10 are written $10^2 = 100$, $10^3 = 1000$, $10^4 = 10\,000$.

Squaring and square rooting are inverse operations, as are cubing and cube rooting. Every positive number has two square roots, one positive and one negative, because $(-5)^2 = 25$ as well as $5^2 = 25$. So $\sqrt{25}$ written with the root sign means the positive root, 5, and the two solutions of $x^2 = 25$ are $x = 5$ and $x = -5$. A negative number has no real square root, since no real number squared is negative. The cube root of a negative number does exist, $\sqrt[3]{-64} = -4$, because $(-4)^3 = -64$.

Inverse operations

Addition and subtraction undo each other, as do multiplication and division. This is the basis for checking a calculation and for solving equations: if $a \div b = c$ then $c \times b = a$. For a chain of operations, undo them in the reverse order. If a number is multiplied by 3 and then 4 is added to give 19, working back gives $19 - 4 = 15$ and then $15 \div 3 = 5$.

Number systems and binary

The number system used every day is base 10, or denary: each place is worth ten times the place to its right, so the place values are $\ldots 1000, 100, 10, 1$. Binary is base 2, where each place is worth twice the place to its right, so the place values from the right are $1, 2, 4, 8, 16, 32, 64, \ldots$ and only the digits 0 and 1 are used. Computers use it because a circuit has two states.

Binary to denary

Add the place values wherever there is a 1.

Denary to binary

Find the largest power of 2 that fits, write a 1 in that place, subtract and continue with the remainder, putting 0 in any place that does not fit. To write 53: $32$ fits, leaving $21$; $16$ fits, leaving $5$; $8$ does not fit; $4$ fits, leaving $1$; $2$ does not fit; $1$ fits.

An alternative is repeated division by 2, reading the remainders from the last to the first. Dividing 53 by 2 gives 26 remainder 1, then 13 remainder 0, 6 remainder 1, 3 remainder 0, 1 remainder 1 and 0 remainder 1, and reading upwards gives $110101$ again.