The structure of the number system: ordering integers and decimals, the four operations with positive and negative numbers, place value, the priority of operations (BIDMAS), and inverse operations.

What this dot point is asking

Edexcel wants you to handle the basics of the number system fluently and, crucially, without a calculator on Paper 1. That means ordering integers, decimals and fractions; adding, subtracting, multiplying and dividing positive and negative numbers; understanding place value; applying the priority of operations; and using inverse operations to check or rearrange. These skills are not glamorous, but arithmetic slips here cost marks across every other topic, so accuracy is the goal.

Ordering numbers and place value

Place value tells you what each digit is worth. In $3.408$ the $4$ is four tenths, the $0$ is zero hundredths and the $8$ is eight thousandths. To order decimals, compare the digits from the left: line up the decimal points and look at the largest place value first. So $0.34$ is bigger than $0.299$ because $3$ tenths beats $2$ tenths, even though $0.299$ looks longer.

A number line is the safest way to order negatives. Numbers increase from left to right, so $-7$ is less than $-3$, and $-3$ is less than $1$. A frequent error is to think $-7 > -3$ because $7 > 3$; on the number line $-7$ sits further left, so it is the smaller number.

The four operations with negatives

The sign rules for multiplying and dividing are worth memorising because they appear everywhere, from substitution to coordinates.

Subtracting a negative is the same as adding: $8 - (-3) = 8 + 3 = 11$. Adding a negative is the same as subtracting: $5 + (-9) = 5 - 9 = -4$. Rewriting "double signs" before you calculate removes most mistakes.

Formal written methods

Paper 1 has no calculator, so you must be fluent in column addition and subtraction, long multiplication and short or long division. For decimals, the key is the decimal point. To add or subtract, line up the decimal points. To multiply, ignore the points, multiply as whole numbers, then count the total decimal places. To divide by a decimal, make the divisor a whole number by multiplying both numbers by a power of ten: $4.8 \div 0.2$ becomes $48 \div 2 = 24$.

The priority of operations (BIDMAS)

When an expression mixes operations, BIDMAS fixes the order so that everyone gets the same answer.

For $20 - 3 \times 4$, the multiplication comes first: $20 - 12 = 8$, not $17 \times 4$. For $(20 - 3) \times 4$, the brackets force the subtraction first: $17 \times 4 = 68$. Indices outrank multiplication, so in $2 \times 3^2$ you square first: $2 \times 9 = 18$.

Inverse operations

Every operation has an inverse that undoes it: addition and subtraction are inverses, and multiplication and division are inverses. Inverses let you check a calculation (if $7 \times 8 = 56$, then $56 \div 8$ should give $7$) and they are the engine behind rearranging equations later in algebra. Squaring and square-rooting are also inverses, which matters for Pythagoras and quadratics.