Knowledge organisers

Number: Cambridge IGCSE Maths (0580) knowledge organiser

Everything to know about number on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Prime number: A whole number with exactly two factors, 1 and itself (1 is not prime).
Significant figures: Count from the first non-zero digit: 0.04056 to 2 s.f. is 0.041.
Standard form: A × 10ⁿ with 1 ≤ A < 10 and n an integer.
Bounds: A value rounded to the nearest u lies between half a unit below and half a unit above it.

Key formulas

Indices	$a^m\times a^n=a^{m+n},$ $a^m\div a^n=a^{m-n},$ $(a^m)^n=a^{mn},$ $a^0=1,$ $a^{-n}=\tfrac1{a^n}$
Fractional indices	$a^{\frac1n}=\sqrt[n]a,$ $a^{\frac mn}=\big(\sqrt[n]a\big)^m$
Standard form	$A\times10^n,$ $1\le A<10,\ n\in\mathbb Z$
Sets	$n(A\cup B)=n(A)+n(B)-n(A\cap B)$
Percentage change	$\frac{\text{new}-\text{original}}{\text{original}}\times100\%$
Reverse percentage	$\text{original}=\frac{\text{new}}{1\pm\frac r{100}}$
Simple interest (rate $r$% a year, $n$ years)	$I=\frac{Prn}{100}$
Compound interest	$A=P\left(1+\tfrac r{100}\right)^n$
Growth and decay	$y=a\times k^n:$ $k>1\text{ growth},$ \(0
Surds	$\sqrt{ab}=\sqrt a\sqrt b,$ $\frac a{\sqrt b}=\frac{a\sqrt b}b,$ $\frac1{c+\sqrt b}=\frac{c-\sqrt b}{c^2-b}$
Recurring decimal: $x=0.\dot4\dot5$, so $100x-x=45$ and $x=\tfrac{45}{99}=\tfrac5{11}$
Bounds: a value to the nearest $u$ lies in $[x-\tfrac u2,\ x+\tfrac u2)$
Bounds in calculations: for the largest $a-b$ or $a\div b$ use the upper bound of $a$ with the lower bound of $b$
Rates	$\text{speed}=\frac{\text{distance}}{\text{time}},$ $\text{density}=\frac{\text{mass}}{\text{volume}},$ $\text{pressure}=\frac{\text{force}}{\text{area}}$

Worked example

A price of $80 is increased by 15%, then the new price is decreased by 15%. Find the final price.

Increase: 80 × 1.15 = 92
Decrease: 92 × 0.85 = 78.2

Answer: $78.20 (not back to $80)

Common mistakes

Rounding in the middle of a calculation
Ignoring the 3 s.f. (1 d.p. for angles) accuracy rule
Standard form answers not in the form A × 10ⁿ
Set notation: counting the overlap twice

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

Complete a short baseline check on number and algebra from earlier years so the teacher can see what each student already knows and plan the first unit.
Add, subtract, multiply and divide decimals without a calculator, such as 0.48 ÷ 0.012, using written methods and place value.
Round numbers to a given number of decimal places or significant figures, and choose an accuracy suited to the context.
Use rates such as density, pressure and population density, rearranging the formulas and stating the correct units.
Use the 24-hour clock, read timetables and solve problems involving time zones and journey times.
Practise standard form and estimation without a calculator, such as dividing numbers in standard form and estimating by rounding.

The printable sheet

Number knowledge organiser for Cambridge IGCSE Maths (0580): one A4 page of key definitions, formulas, a worked example and common mistakes — Number knowledge organiser (Cambridge IGCSE Maths (0580)), A4. Download the PDF.

Revise it next

Other Cambridge IGCSE Maths (0580) topics: Algebra and graphs · Coordinate geometry · Geometry · Mensuration · Trigonometry · Transformations and vectors · Probability · Statistics · All Cambridge IGCSE Maths (0580) organisers

Indices	\(a^m\times a^n=a^{m+n},\) \(a^m\div a^n=a^{m-n},\) \((a^m)^n=a^{mn},\) \(a^0=1,\) \(a^{-n}=\tfrac1{a^n}\)
Fractional indices	\(a^{\frac1n}=\sqrt[n]a,\) \(a^{\frac mn}=\big(\sqrt[n]a\big)^m\)
Standard form	\(A\times10^n,\) \(1\le A<10,\ n\in\mathbb Z\)
Sets	\(n(A\cup B)=n(A)+n(B)-n(A\cap B)\)
Percentage change	\(\frac{\text{new}-\text{original}}{\text{original}}\times100\%\)
Reverse percentage	\(\text{original}=\frac{\text{new}}{1\pm\frac r{100}}\)
Simple interest (rate \(r\)% a year, \(n\) years)	\(I=\frac{Prn}{100}\)
Compound interest	\(A=P\left(1+\tfrac r{100}\right)^n\)
Growth and decay	\(y=a\times k^n:\) \(k>1\text{ growth},\) \(0
Surds	\(\sqrt{ab}=\sqrt a\sqrt b,\) \(\frac a{\sqrt b}=\frac{a\sqrt b}b,\) \(\frac1{c+\sqrt b}=\frac{c-\sqrt b}{c^2-b}\)
Recurring decimal: \(x=0.\dot4\dot5\), so \(100x-x=45\) and \(x=\tfrac{45}{99}=\tfrac5{11}\)
Bounds: a value to the nearest \(u\) lies in \([x-\tfrac u2,\ x+\tfrac u2)\)
Bounds in calculations: for the largest \(a-b\) or \(a\div b\) use the upper bound of \(a\) with the lower bound of \(b\)
Rates	\(\text{speed}=\frac{\text{distance}}{\text{time}},\) \(\text{density}=\frac{\text{mass}}{\text{volume}},\) \(\text{pressure}=\frac{\text{force}}{\text{area}}\)