Maths reference: Number

A whole number, zero, or a negative whole number.

A number below zero, to the left of $0$ on a number line: five below zero.

The wide end of the sign faces the greater number. Further left on the line is less.

The size of a number without its sign. Among negatives, the one further from zero is the lesser: $-10 < -1$.

A number times itself: the area of a square with that side. The perfect squares to 144: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.

The number that times itself makes this one: the side of a square with that area. Not half: $\sqrt{36} \ne 18$.

The base (2) is multiplied; the exponent (5) counts how many times. $2^5$ is not $2 \times 5$.

A number with exactly two factors, 1 and itself. 1 is not prime.

A whole number and a fraction. A negative one is both parts below zero: one and a half below zero.

Each place value column is a power of 10. The exponent counts the zeros, and the ones column is $10^0 = 1$.

A number as the sum of each digit times its place value. A column holding 0 adds nothing, so leave it out.

The number of digits after the decimal point. To 2 decimal places means to the nearest hundredth.

A quick, close answer from rounded numbers. Round to the first digit and keep the size; then compare with the exact answer.

Compares quantities in a stated order. The first thing named is the first number.

Part to whole adds the parts: $2 + 3 = 5$. The part-to-whole ratio $2 : 5$ matches the fraction $\tfrac{2}{5}$.

Multiply or divide both terms by the same number. Adding the same number to both changes the ratio.

Divide both terms by their highest common factor (here 6). No number above 1 divides both.

A ratio of the drawing (or model) to the real thing. Each 1 unit on the drawing is 50 units in real life.

The first step in modelling: say what you know, what you need, and how they are linked. Then calculate.

One change after another: multiply the multipliers. Up 20% then down 20% is × 0.96.

The marginal rate is the rate on the last dollar; the effective rate is total tax ÷ income.

A rate compares quantities with different units (km per h, $ per kg); a ratio compares the same kind. 1 km/h = 1000 m ÷ 3600 s: divide by 3.6 for m/s.

Write the units in: km ÷ km/h = h. Then check the size makes sense.

Every model assumes something (steady speed, no fees, no starting charge). When it disagrees with real life, name the assumption and refine the model.

Start at the first number. Adding a positive moves right; adding a negative moves left.

Subtracting moves the opposite way to adding: subtracting a positive moves left, past zero if it has to.

Adding $-3$ would move left 3, so subtracting it moves right 3. It gives the same as adding $3$.

Same signs give a positive, different signs a negative. On the line: $a$ jumps of $b$; a negative number of jumps goes the opposite way.

The same sign rule as multiplying, because dividing undoes it: check $3 \times (-4) = -12$.

Find the perfect squares either side and take their roots. Square decimals to get closer: $6.6^2 = 43.56$, so $\sqrt{43} \approx 6.6$.

Split into factor pairs until every branch is a prime, then write repeated primes with exponents. Every tree gives the same primes.

HCF: the primes both share. LCM: all the primes, shared ones once. Sharing into equal groups needs the HCF; things lining up again need the LCM.

Multiply (or divide) the top and the bottom by the same number: the same amount, cut into more (or fewer) parts. Adding the same number to both changes the amount.

Divide the top and the bottom by their highest common factor. In simplest form (lowest terms) they share no factor but 1.

Percent means out of 100: make the fraction out of 100, or work out the decimal and multiply by 100.

Divide the top by the bottom. Some never end: $\tfrac{1}{3} = 0.333\ldots$

Decimal to percentage: × 100 (0.5 is 50 hundredths, not 5%). Percentage to decimal: ÷ 100.

Put it over 100, then simplify.

Cut both into the same parts first (a common denominator; the LCM is the lowest), then add the tops. Never add the bottoms.

Multiply the tops and the bottoms, then simplify. Multiplying by a fraction less than 1 makes smaller.

How many fit? Dividing by a fraction is multiplying by it flipped (top and bottom swapped): $\tfrac{2}{3} \div \tfrac{1}{6} = \tfrac{2}{3} \times 6 = 4$.

Of means multiply. Mentally: 10% is a tenth (8), 5% is half of that (4).

15% off leaves 85%: × 0.85. 15% on makes 115%: × 1.15. For what percentage, divide and × 100: $18 \div 24 = 75\%$.

Change the order ($4 \times 2.7 \times 25 = 100 \times 2.7$), change the form (0.75 = 3/4), or split ($12 \times 2.5 = 24 + 6$). Then estimate to check.

Split each unit into equal parts and count from zero. A negative counts left, and the part counts further from zero, not back towards it.

The integer sign rules hold for every fraction and decimal. Sign first, then size.

Multiply in any order, pairing friendly numbers; or split a number: $6 \times (-2.5) = 6 \times (-2) + 6 \times (-0.5)$.

Look at the digit straight after the place. 5 or more rounds up; 4 or less leaves the digit. Halfway rounds up.

Round each number to its first digit, keep its size, calculate, then compare with the exact answer.

Add the parts, divide the total by the parts to find one part, then multiply. The shares add back to the total.

A percentage profit or change is out of the original amount, the cost, not the selling price.

Each percentage is of a different amount, so they cannot be added: 20% off then 10% off is not 30% off.