Edexcel GCSE Mathematics

Place value and ordering

Each digit in a number is worth ten times the digit to its right, so in $4\,062.35$ the 6 is worth 60, the 3 is worth $\frac{3}{10}$ and the 5 is worth $\frac{5}{100}$. A zero holds a place open without adding anything. Place value is what lets you multiply and divide by 10, 100 and 1000 by shifting digits, and what lets you compare decimals: $0.09$ is less than $0.4$ because $0.4$ has 4 tenths and $0.09$ has none.

To order decimals, write them with the same number of decimal places (pad with zeros) and compare digit by digit from the left. To order fractions, give them a common denominator or convert them to decimals. On a number line the smaller number is always to the left, so $-7 < -2$, even though 7 is bigger than 2.

The inequality symbols are $<$ (less than), $>$ (greater than), $\le$ (less than or equal to) and $\ge$ (greater than or equal to), and $\ne$ means not equal to. The open end of the symbol faces the bigger number. For example $-3 \le n < 2$ means $n$ can be $-3$ but must be smaller than $2$, so the integers that fit are $-3, -2, -1, 0, 1$.

Negative numbers

Adding a negative number is the same as subtracting the positive, and subtracting a negative is the same as adding the positive.

When multiplying or dividing, two numbers with the same sign give a positive answer and two with different signs give a negative one. So $(-4) \times (-6) = 24$, $(-4) \times 6 = -24$ and $(-20) \div (-5) = 4$.

Written methods

Addition and subtraction work column by column with digits of equal place value lined up, carrying or exchanging as needed. For decimals, line up the decimal points and fill any gap with a zero, so that $12.4 - 3.85$ becomes $12.40 - 3.85 = 8.55$.

Long multiplication splits one number into its place values. For $47 \times 36$:

To multiply decimals, ignore the decimal 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 together. For $0.7 \times 0.24$, work out $7 \times 24 = 168$; there are 1 + 2 = 3 decimal places, so the answer is $0.168$.

To divide by a decimal, multiply both numbers by the same power of 10 until the divisor is a whole number. So $4.5 \div 0.15$ is the same as $450 \div 15 = 30$.

Short and long division both find how many times the divisor goes into each part of the dividend. A remainder can be written as a whole number, as a fraction of the divisor, or as a decimal, depending on what the question needs. $200 \div 16 = 12$ remainder $8$, which is $12\frac{1}{2}$ or $12.5$.

Order of operations

Calculations are done in a fixed order: brackets first, then powers and roots, then multiplication and division from left to right, then addition and subtraction from left to right. Multiplication and division have equal priority, and so do addition and subtraction, which is why left to right matters.

A fraction bar acts as a bracket around the numerator and around the denominator, and a root sign covers everything under it. $\dfrac{12 + 6}{2 + 1} = \dfrac{18}{3} = 6$, not $12 + 3 + 1$. A calculator follows the same order, but it only sees the brackets you type, so the numerator and denominator of a fraction each need their own brackets when you use the division key.

A negative number squared needs brackets to be squared as a whole: $(-3)^2 = 9$, whereas $-3^2$ means $-(3^2) = -9$.

The reciprocal of a number is the number that multiplies with it to give 1. The reciprocal of 5 is $\frac{1}{5}$, of $\frac{2}{3}$ is $\frac{3}{2}$, and of 0.25 is 4. Zero has no reciprocal.

Inverse operations

Addition and subtraction undo each other, and so do multiplication and division. Squaring and taking the square root undo each other for positive numbers, as do cubing and taking the cube root. Inverse operations are used to check an answer, to solve equations (work backwards through the operations in reverse order) and to simplify. Cancelling is the inverse at work: in $\dfrac{7 \times 15}{15}$ the multiplication by 15 and the division by 15 cancel, leaving 7. In the same way, since $23 \times 48 = 1104$, it follows that $1104 \div 48 = 23$ and $1104 \div 23 = 48$.

Fractions

A fraction shows a part of a whole, and an improper fraction such as $\frac{17}{5}$ has a numerator larger than its denominator. It equals the mixed number $3\frac{2}{5}$, since $17 = 3 \times 5 + 2$.

Equivalent fractions come from multiplying or dividing the numerator and denominator by the same number. A fraction is in its simplest form when the numerator and denominator have no common factor other than 1.

To add or subtract, write both fractions over a common denominator, preferably the lowest common multiple of the two denominators, and add or subtract the numerators.

To multiply, multiply the numerators and multiply the denominators, cancelling common factors first if you can: $\dfrac{4}{9} \times \dfrac{3}{10} = \dfrac{12}{90} = \dfrac{2}{15}$.

To divide by a fraction, multiply by its reciprocal: $\dfrac{3}{4} \div \dfrac{9}{10} = \dfrac{3}{4} \times \dfrac{10}{9} = \dfrac{30}{36} = \dfrac{5}{6}$.

With mixed numbers, convert them to improper fractions before multiplying or dividing. For $2\frac{1}{3} \times 1\frac{1}{2}$ write $\frac{7}{3} \times \frac{3}{2} = \frac{21}{6} = 3\frac{1}{2}$. For addition and subtraction of mixed numbers either convert the same way or add the whole numbers and the fractions separately, taking care when a subtraction needs an exchange. The rules for signs apply to fractions in the same way as to whole numbers, so $-\frac{1}{2} \times \frac{2}{5} = -\frac{1}{5}$.