Ordering, standard form, estimation, bounds, indices, surds

Cambridge IGCSE Mathematics 0580 – Syllabus Notes (Core & Extended)

Learning Objectives (Number Strand)

• Classify and work with the different number sets used in the syllabus.
• Use set notation and Venn diagrams to describe relationships between number groups.
• Order integers, fractions, decimals, percentages and surds accurately.
• Write numbers in standard (scientific) form and perform the four operations with them.
• Make sensible estimates, apply rounding, and state upper‑ and lower‑bounds (including for surds).
• Convert between fractions, decimals and percentages and apply BODMAS.
• Apply the laws of indices (positive, zero, negative and fractional powers) to simplify expressions.
• Extract and use common powers and roots (square, cube, 4th‑root, …).
• Manipulate surds – simplify, multiply, divide, rationalise denominators and find bounds.

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1. Types of Numbers

Number type	Definition	Examples
Natural numbers (N)	Positive integers (counting numbers)	1, 2, 3, …
Integers (Z)	…, −2, −1, 0, 1, 2, …	−5, 0, 12
Prime numbers	Integers > 1 with exactly two distinct positive factors	2, 3, 5, 7, 11,…
Square numbers	Numbers of the form $n^{2}$	1, 4, 9, 16, 25,…
Cube numbers	Numbers of the form $n^{3}$	1, 8, 27, 64,…
Rational numbers (Q)	Can be written as $\dfrac{p}{q}$ with $qeq0$ (terminating or repeating decimals)	$\dfrac{3}{4}=0.75$, $-2$, $0.\overline{3}$
Irrational numbers	Cannot be expressed as a fraction; decimal expansion is non‑terminating & non‑repeating	$\sqrt{2},\;\pi$
Surds (Extended only)	Irrational roots that cannot be simplified to a terminating decimal	$\sqrt{3},\;\sqrt[3]{7}$

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2. Sets & Venn Diagrams

• Notation: $A$, $B$, $A\cup B$ (union), $A\cap B$ (intersection), $A^{\prime}$ (complement).
• Universal set ($U$) – the set of all numbers under consideration (e.g. all integers from $-10$ to $10$).

Example – Squares and Cubes (integers 1–20)

Let $S$ be the set of perfect squares, $C$ the set of perfect cubes.

• $S=\{1,4,9,16\}$
• $C=\{1,8\}$
• Intersection $S\cap C=\{1\}$.

A two‑circle Venn diagram would place 1 in the overlap, 4, 9, 16 only in $S$, and 8 only in $C$.

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3. Ordering Numbers

Arrange from smallest to largest by comparing the most significant part first.

1. Integers – compare directly.
2. Fractions – find a common denominator or convert to decimals.
3. Decimals – compare digit‑by‑digit from left to right.
4. Percentages – treat as decimals (e.g. $45\% = 0.45$).
5. Surds – approximate to 2–3 dp or use bounds (see §5).

Worked Example

Arrange in ascending order:

$$-3.2,\;\frac{7}{4},\;2.5,\;\sqrt{5},\;-\frac{9}{2},\;45\%$$

• $\frac{7}{4}=1.75$
• $\sqrt{5}\approx2.236$
• $-\frac{9}{2}=-4.5$
• $45\% =0.45$

Ordered list: $-\dfrac{9}{2},\;-3.2,\;0.45,\;1.75,\;2.236,\;2.5$.

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4. Standard Form (Scientific Notation)

Any non‑zero number can be written as

$$a\times10^{n}\qquad\text{where }1\le|a|<10,\;n\in\mathbb Z.$$

Conversion Table

Ordinary notation	Standard form
0.00034	$3.4\times10^{-4}$
5 200 000	$5.2\times10^{6}$
-0.00789	$-7.89\times10^{-3}$
9.3×10⁸	$9.3\times10^{8}$ (already in standard form)

Operations

• Multiplication: $(a\times10^{m})(b\times10^{n})=(ab)\times10^{m+n}$
• Division: $\dfrac{a\times10^{m}}{b\times10^{n}}=\dfrac{ab^{-1}}{1}\times10^{m-n}$ (simplify the coefficient then adjust).
• Addition / Subtraction: first ensure the same power of ten, add/subtract the coefficients, then re‑express in standard form.

Example – Multiplying in Standard Form

$$ (4.2\times10^{3})\times(3.5\times10^{-2}) = (4.2\times3.5)\times10^{3-2}=14.7\times10^{1}=1.47\times10^{2} $$

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5. Estimation, Rounding & Bounds

5.1 Estimating Sums and Products

Round each term to a convenient value, perform the operation, then state a reasonable range for the exact answer.

Example – Estimate $23.7\times4.9$

Round to $24\times5=120$. The exact product is $116.13$, which lies between $115$ and $125$.

5.2 Upper and Lower Bounds for Decimal Answers

• If a value is given to $d$ decimal places, the lower bound = value $-0.5\times10^{-d}$ (inclusive).
• Upper bound = value $+0.5\times10^{-d}$ (exclusive).

Given value	Lower bound	Upper bound
3.45 (2 dp)	3.445 (inclusive)	3.455 (exclusive)
0.7 (1 dp)	0.65	0.75
$\sqrt{50}$ (to 2 dp)	7.07	7.08

5.3 Bounds for Surds (Extended)

Locate consecutive perfect squares $k^{2}$ and $(k+1)^{2}$ such that $k^{2}

Example – Bounds for $\sqrt{72}$

$8^{2}=64$, $9^{2}=81\;\Rightarrow\;8<\sqrt{72}<9$.

Because $8.5^{2}=72.25$, a tighter bound is $8<\sqrt{72}<8.5$.

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6. Fractions, Decimals & Percentages

6.1 Conversion Table

Fraction	Decimal	Percentage
$\dfrac{3}{4}$	0.75	75 %
$\dfrac{7}{20}$	0.35	35 %
$\dfrac{5}{8}$	0.625	62.5 %
0.28	0.28	28 %
42 %	0.42	42 %

6.2 Operations

• Add / Subtract: use a common denominator or work in decimals.
• Multiply: multiply numerators and denominators directly; for decimals, treat as whole numbers then place the decimal point.
• Divide: multiply by the reciprocal; for decimals, move the decimal point in the divisor to make it a whole number and do the same to the dividend.

Worked Example – $\dfrac{3}{5}+0.4$

Convert $0.4$ to a fraction: $0.4=\dfrac{4}{10}=\dfrac{2}{5}$. Then

$$\dfrac{3}{5}+\dfrac{2}{5}=\dfrac{5}{5}=1.$$

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7. The Four Operations & BODMAS

Order of operations (BODMAS): Brackets → Orders (indices, roots) → Division/Multiplication → Addition/Subtraction. Multiplication and division are performed left‑to‑right, as are addition and subtraction.

Example – Evaluate $6+2\times(3^{2}-5)\div4$

1. Brackets: $3^{2}=9\;\Rightarrow\;9-5=4$.
2. Orders: already done.
3. Multiplication/Division (left‑to‑right): $2\times4=8$, then $8\div4=2$.
4. Addition: $6+2=8$.

Result: $8$.

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8. Indices (Powers)

8.1 Types of Indices

• Positive integer: $a^{n}$ (e.g. $2^{3}=8$).
• Zero: $a^{0}=1$ (for $aeq0$).
• Negative integer: $a^{-n}= \dfrac{1}{a^{n}}$.
• Fractional: $a^{\frac{1}{n}}=\sqrt[n]{a}$, $a^{\frac{m}{n}}=\sqrt[n]{a^{m}}$.

8.2 Laws of Indices

$$\begin{aligned} a^{m}\times a^{n}&=a^{m+n}\\[2mm] \dfrac{a^{m}}{a^{n}}&=a^{m-n}\\[2mm] (a^{m})^{n}&=a^{mn}\\[2mm] (ab)^{n}&=a^{n}b^{n}\\[2mm] a^{-n}&=\dfrac{1}{a^{n}}\\[2mm] a^{\frac{1}{n}}&=\sqrt[n]{a} \end{aligned}$$

8.3 Power Table (Core)

Base $a$	$a^{2}$	$a^{3}$	$a^{4}$	$a^{-1}$	$\sqrt{a}$
2	4	8	16	0.5	$\sqrt{2}\approx1.414$
5	25	125	625	0.2	$\sqrt{5}\approx2.236$
10	100	1 000	10 000	0.1	3.162

8.4 Extended Example – Simplify $\displaystyle \frac{2^{5}\times3^{-2}}{(2\times3)^{2}}$

\[ \frac{2^{5}\times3^{-2}}{(2\times3)^{2}} =\frac{2^{5}\times3^{-2}}{2^{2}\times3^{2}} =2^{5-2}\times3^{-2-2}=2^{3}\times3^{-4} =8\times\frac{1}{3^{4}}=8\times\frac{1}{81}=\frac{8}{81}. \]

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9. Surds (Extended)

9.1 Definition & Simplification

A surd is an irrational root that cannot be expressed as a terminating decimal. Simplify by extracting perfect powers.

Example – Simplify $\sqrt{72}$

\[ \sqrt{72}=\sqrt{36\times2}=6\sqrt{2}. \]

9.2 Operations with Surds

• Multiplication: $\sqrt{a}\times\sqrt{b}=\sqrt{ab}$.
• Division: $\dfrac{\sqrt{a}}{\sqrt{b}}=\sqrt{\dfrac{a}{b}}$ (rationalise if required).
• Rationalising denominators: multiply numerator and denominator by the appropriate surd.

Example – Rationalise $\displaystyle \frac{5}{\sqrt{3}}$

\[ \frac{5}{\sqrt{3}}=\frac{5}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}=\frac{5\sqrt{3}}{3}. \]

9.3 Bounds for Surds (see §5.3)

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Other Syllabus Strands – Quick Reference (Core & Extended)

Strand	Core Topics (C)	Extended Topics (E)
Algebra & Graphs	C2.1–C2.12 (expressions, linear equations, simultaneous equations, quadratic equations, algebraic fractions, sequences, graphs of linear & quadratic functions)	E2.1–E2.13 (including completing the square, factorising quadratics, solving higher‑order equations, transformation of graphs)
Coordinate Geometry	C3.1–C3.7 (plotting points, gradient, equation of a line, parallel/perpendicular, mid‑point, distance)	E3.1–E3.7 (including using the section formula, equations of circles)
Geometry	C4.1–C4.8 (properties of shapes, constructions, similarity, symmetry, circle theorems, bearings)	E4.1–E4.3 (including transformations, loci, advanced circle theorems)
Mensuration	C5.1–C5.5 (area & perimeter of plane figures, surface area & volume of solids, compound shapes)	E5.1–E5.5 (including nets, surface area of composite solids)
Trigonometry	C6.1–C6.2 (Pythagoras, sine, cosine, tangent for right‑angled triangles)	E6.1–E6.2 (including solving oblique triangles using the sine & cosine rules)

Use the tables above to check which topics you need to study for your specific examination tier (Core or Extended). The detailed notes in sections 1–9 cover the entire Number strand, which is a prerequisite for all other areas.