Order positive and negative integers, decimals and fractions; use the four operations and the correct order of operations (BIDMAS), including with negatives.

What this dot point is asking

OCR groups the structure of the number system and calculation under references N1 and N2. You must order positive and negative integers, decimals and fractions, and apply the four operations with the correct order of operations, including when negative numbers are involved. These skills sit underneath every other topic in the qualification: a sign slip or a BIDMAS error in the first line of working carries through the rest of a question, so accuracy here protects marks everywhere. This content is assessed on every paper, and because it appears on the non-calculator Paper 2 (or Paper 5 at Higher) you must be fluent by hand.

Types of number and ordering

The number line runs from large negatives through zero to large positives. Integers are whole numbers (positive, negative or zero); the rationals include every fraction and terminating or recurring decimal; the reals fill in the irrationals such as $\sqrt{2}$ and $\pi$. For ordering you rarely need the names, but you do need a reliable comparison method.

To order a mixed list, put everything in the same form. Comparing $\tfrac{3}{4}$, $0.7$ and $\tfrac{5}{8}$ is easiest as decimals: $0.75$, $0.7$ and $0.625$, so the order from smallest is $\tfrac{5}{8}, 0.7, \tfrac{3}{4}$. With negatives, remember the line reverses your intuition: $-9 < -2$ because $-9$ is further from zero on the negative side. A common exam phrasing asks you to order temperatures or bank balances; treat "coldest" or "most overdrawn" as "smallest".

The four operations with directed numbers

Addition and subtraction of directed numbers are easiest pictured as moves on the number line: adding a positive moves right, adding a negative moves left, and subtracting reverses the direction of the move.

For example $-5 - (-8) = -5 + 8 = 3$, and $-4 \times 6 = -24$, while $\dfrac{-30}{-5} = 6$. The trap is mixing up the rule for adding signs with the rule for multiplying signs: $-3 + (-4) = -7$ (a subtraction), but $-3 \times -4 = 12$ (a positive product).

The order of operations (BIDMAS)

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

So $5 + 2 \times 3^2 = 5 + 2 \times 9 = 5 + 18 = 23$: the index first, then the multiplication, then the addition. With a fraction bar, $\dfrac{8 + 4}{2 + 1} = \dfrac{12}{3} = 4$, because the bar groups the top and the bottom before you divide.

Why this underpins everything

Every later topic assumes you can calculate accurately with signs and operations: substituting a negative into a formula, simplifying an algebraic expression, finding a gradient between two points, or evaluating the discriminant $b^2 - 4ac$ when $b$ is negative. OCR rewards method, so even when the arithmetic is the hard part, setting out each step keeps method marks secure. On the non-calculator paper especially, written multiplication, short and long division, and careful column addition are the tools that turn a correct plan into a correct answer.