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NUMBER LAB
Learn it by playing it
Four short missions on live number lines. Grab a dial, watch the number move, and figure out the rule. When a mission glows green, read why it worked.
1234
Mission 1 of 4
Make forty-five thousand six hundred
Make 45,600. Adjust the front number and power of 10; watch the marker.
1 x 10^0 = 1
Before you touch the dials: 4.56 x 10^4 is how many times bigger than 4.56 x 10^2?
Currently 1 - too small. Push the mantissa up, or add a power of ten.
Study notes
Types of number C1.1 / E1.1
Natural numbers are the counting numbers (1, 2, 3, ...); integers add zero and the negatives.
A prime number has exactly two factors: 1 and itself. 1 is not prime.
Every integer greater than 1 breaks down uniquely into prime factors (the Fundamental Theorem of Arithmetic). Write 72 = 2³ × 3².
The HCF (highest common factor) of two numbers is the largest shared factor; the LCM (lowest common multiple) is the smallest shared multiple. Prime factorisation finds both: take shared primes at their lowest powers for the HCF, all primes at their highest powers for the LCM.
Rational numbers can be written as a fraction of two integers (this includes terminating and recurring decimals); irrational numbers (like √2 or π) cannot.
The reciprocal of a non-zero number n is 1/n; a non-zero number multiplied by its reciprocal is 1, and zero has no reciprocal.
Sets C1.2 / E1.2
A set is a collection of distinct items, written in curly brackets, e.g. A = {2, 4, 6}.
n(A) means the number of elements in set A.
The universal set contains everything under discussion; the empty set contains nothing.
The complement of A is everything in the universal set that is not in A.
Union gathers every element in either set; intersection keeps only elements in both.
Venn diagrams show these regions; shading questions are answered region by region.
Powers, roots and indices C1.3, C1.7 / E1.3, E1.7
Squares (1, 4, 9, 16, ...), cubes (1, 8, 27, ...) and higher powers come from repeated multiplication.
Index laws: when multiplying powers of the same base, add the indices; when dividing, subtract them; a power of a power multiplies the indices.
A zero index gives 1 for a non-zero base; a negative index means the reciprocal, so a⁻² = 1/a² for a ≠ 0.
Fractional indices are roots of a positive base: a power of 1/2 is the square root, 1/3 is the cube root, and 2/3 is the cube root then squared.
Estimate roots by trapping them between known squares or cubes before reaching for a calculator.
Fractions, decimals and percentages C1.4, C1.6 / E1.4, E1.6
Convert freely between the three forms: a percentage is a fraction out of 100; divide numerator by denominator for the decimal.
To add or subtract fractions, use a common denominator; to multiply, multiply top and bottom; to divide, multiply by the reciprocal.
Mixed numbers convert to improper fractions before calculating, then back if the question asks.
Recurring decimals convert to fractions algebraically: set x equal to the decimal, multiply to line up the recurring part, subtract and solve.
Order the four operations with brackets first, then indices, then multiplication/division, then addition/subtraction.
Ordering and estimation C1.5, C1.9 / E1.5, E1.9
Order quantities by magnitude - including fractions, decimals, percentages, standard form and surds - by converting to a common form first.
Use the symbols =, not-equal, less-than, greater-than and their or-equal forms correctly.
Estimate by rounding every value to one significant figure before calculating; state whether your estimate over- or under-estimates when the rounding direction is clear.
Round to a given number of decimal places or significant figures; counting starts at the first non-zero digit.
The four operations C1.6 / E1.6
Follow the agreed order of operations: brackets first, then powers and roots, then multiplication and division, then addition and subtraction.
Adding or subtracting a negative flips the sign: 5 - (-3) = 8. Two negatives multiplied or divided give a positive; one of each gives a negative.
Add and subtract fractions over a common denominator; multiply straight across; to divide, multiply by the reciprocal.
Convert mixed numbers to improper fractions before multiplying or dividing them.
Sanity-check each calculation against a quick estimate so a slipped order of operations does not survive.
Standard form C1.8 / E1.8
Standard form writes a number as A × 10ⁿ where 1 ≤ A < 10 and n is an integer.
Large numbers take positive n (4 500 000 = 4.5 × 10⁶); small numbers take negative n (0.00032 = 3.2 × 10⁻⁴).
To multiply or divide in standard form, handle the A parts and the powers of ten separately, then renormalise so A stays in range.
To add or subtract, first rewrite both numbers with the same power of ten.
Convert back to ordinary form by moving the decimal point n places.
Limits of accuracy (bounds) C1.10 / E1.10
A rounded value sits inside an interval half a unit wide on either side. A length of 8.4 cm rounded to 1 decimal place lies between 8.35 cm (lower bound) and 8.45 cm (upper bound).
The upper bound itself is not included in the interval, but it is still the value used in bound calculations.
For the largest result of a calculation, combine bounds to maximise it (e.g. biggest numerator, smallest denominator); for the smallest result, do the reverse.
Always write bounds to one more decimal place than the rounded value when needed, and keep full precision until the final answer.
Ratio, proportion and rates C1.11, C1.12 / E1.11, E1.12
A ratio compares parts; divide it by common factors to simplify, just like a fraction.
To share an amount in a ratio, add the parts, find the value of one part, then scale.
Increase or divide a quantity in a given ratio by multiplying by the ratio as a fraction.
Rates compare different units - speed, density, flow. Average speed = total distance / total time; unit rates come from dividing by the quantity of the second unit.
Direct proportion means one quantity is a fixed multiple of the other; use the constant of proportionality to move between them.
Percentages C1.13 / E1.13
Find a percentage of an amount by multiplying by the percentage as a decimal.
Percentage change = (change / original) × 100, measured against the original value, not the new one.
To increase by r percent, multiply by (1 + r/100); to decrease, multiply by (1 - r/100). Multipliers make repeated change easy.
Reverse percentages recover the original: divide the final value by the multiplier, never by subtracting the percentage.
Simple interest adds the same amount each period: I = P × r × t.
Using a calculator C1.14 / E1.14
Enter long calculations in one chain with brackets instead of retyping rounded intermediate values.
Know the keys for powers, roots, fractions and standard form, and check the display mode before trusting an answer.
Estimate the order of magnitude first so a mis-keyed bracket stands out.
Keep full precision until the final line, then round to the accuracy the question asks for.
Time and money C1.15, C1.16 / E1.15, E1.16
Work in 24-hour time or convert carefully; add and subtract times in hours and minutes, carrying in base 60.
Read timetables by finding the row and column for the two stops and subtracting the times.
Money questions combine unit conversion (for example rupees to paise) with the four operations; give answers to two decimal places.
Wages, bills, discounts and profit/loss are percentage and rate problems in disguise - identify the multiplier or the rate first.
Exponential growth and decay (Extended) E1.17
Exponential change multiplies by the same factor every period: final = initial times the multiplier raised to the number of periods.
Growth uses a multiplier above 1 (a 5 percent increase means 1.05); decay uses a multiplier below 1 (a 12 percent decrease means 0.88).
Depreciation, population change and compound interest are all this model; compound interest over t years is P × (1 + r/100)ᵗ.
To find the rate from an initial and final value, divide, take the t-th root, and read the multiplier.
Surds (Extended) E1.18
A surd is an exact irrational root, like the square root of 5. Keep surds exact instead of rounding.
Simplify by pulling out square factors: √50 = √(25 × 2) = 5√2.
Add and subtract only like surds; multiply surds by multiplying the numbers under one root.
Rationalise a denominator of the form √a by multiplying top and bottom by √a; for a + √b, multiply by its conjugate a − √b to clear the root from the denominator.
EduTribe study notes - original material aligned to the Cambridge IGCSE Mathematics 0580 syllabus (2025-2027). Automated checks completed; not yet reviewed by an educator.