SPI Mathematics · Free notes

Number System: SPI Mathematics

Number System has 47 questions in the SPI bank, and six of them are from the 2022 paper. None is HARD: 34 are EASY and 13 are MODERATE, and most take one step once you know the fact behind them, such as what a digit is worth in its place, how a number splits into primes, or why the root of a number that is not a perfect square is irrational. Work the pages in order, because HCF and LCM are built from prime factors and fraction sums use the LCM.

The handout has every subtopic, worked example, formula and trap with answers shown. The formula sheet is the chapter’s formulas and traps alone, on one page.

SPI home All formulas on one page

Subtopic notes

Formula & revision sheet

10 formulas · 3 reference tables · 30 gotchas across all subtopics — the exam-eve cheat-sheet

Number Types and Place Value

Formulas (2)

Place value and face value · Place value $\text{place value} = \text{digit} \times \text{value of its place}$
Counting digits and numbers made from digits · Sum of all numbers from three different non-zero digits $\text{sum} = 2 \times (a + b + c) \times 111 \qquad \text{numbers from } m \text{ to } n = n - m + 1$

Reference tables (1)

Natural numbers, whole numbers, integers, even and odd6 rows

Family	What it holds	Examples
Natural numbers	The counting numbers, starting at 1	1, 2, 3, 4, ...
Whole numbers	The natural numbers and 0	0, 1, 2, 3, ... 0 is the smallest whole number. There is no smallest integer.
Integers	The whole numbers and their negatives	..., $-2$ , $-1$ , 0, 1, 2, ...
Even integers	Numbers of the form $2m$	$-4$ , 0, 6, 10
Odd integers	Numbers of the form $2m + 1$	$-3$ , 1, 7, 11
Powers of $-1$	1 for an even power, $-1$ for an odd power	$(-1)^4 = 1$ , $(-1)^7 = -1$

Each family contains the one above it: natural numbers are whole numbers, and whole numbers are integers.

Watch out for (7)

Calling 0 a natural number
→ Natural numbers, whole numbers, integers, even and odd
Losing track of alternating signs
→ Natural numbers, whole numbers, integers, even and odd
Confusing place value with face value
→ Place value and face value
Counting places from the left
→ Place value and face value
The smallest five-digit number
→ Place value and face value
Counting a range one short
→ Counting digits and numbers made from digits
Using the 111 rule when one digit is 0
→ Counting digits and numbers made from digits

Divisibility, Primes and Factorisation

Formulas (1)

Prime factorisation · Prime factorisation and end zeros $n = p_1^{a_1} \times p_2^{a_2} \times \cdots \times p_k^{a_k} \qquad \text{zeros at the end of } 2^a \times 5^b \times \ldots = \text{the smaller of } a, b$

Reference tables (1)

Divisibility rules8 rows

Divisor	Rule	Example
2	Last digit is 0, 2, 4, 6 or 8	3576
3	Sum of the digits is divisible by 3	741: $7 + 4 + 1 = 12$
4	Last two digits form a number divisible by 4	5316: 16
5	Last digit is 0 or 5	8935
6	Divisible by both 2 and 3	912: even, $9 + 1 + 2 = 12$
8	Last three digits form a number divisible by 8	71240: 240
9	Sum of the digits is divisible by 9	6534: $6 + 5 + 3 + 4 = 18$
11	Alternating sum of the digits is 0 or a multiple of 11	3729: $3 - 7 + 2 - 9 = -11$ Add and subtract the digits in turn. The plain digit sum says nothing about 11.

Each example in the last column passes its rule.

Watch out for (7)

Taking away multiples of the product, not the LCM
→ Divisibility rules
Testing 6 with one rule
→ Divisibility rules
Calling 1 a prime
→ Prime and composite numbers
Numbers that look prime
→ Prime and composite numbers
Stopping at a composite piece
→ Prime factorisation
Adding the powers of 2 and 5 for end zeros
→ Prime factorisation
Squaring the divisor
→ Prime factorisation

HCF and LCM

Formulas (3)

HCF and LCM from prime factors · HCF and LCM by powers $\text{HCF} = \prod p^{\text{lowest power}} \qquad \text{LCM} = \prod p^{\text{highest power}}$
Euclid's division and the largest number that divides · Euclid's division $a = bq + r, \quad 0 \le r < b \qquad \text{HCF}(a, b) = \text{HCF}(b, r)$
HCF times LCM equals the product · Product of two numbers $\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$

Watch out for (5)

Putting a prime in the HCF that one number lacks
→ HCF and LCM from prime factors
Picking a common multiple that is not the least
→ HCF and LCM from prime factors
Taking the HCF before removing the remainder
→ Euclid's division and the largest number that divides
An answer smaller than the remainder
→ Euclid's division and the largest number that divides
Using the product rule for three numbers
→ HCF times LCM equals the product

Fractions and Rational Numbers

Formulas (3)

Rational numbers and comparing them · Rational number $\frac{p}{q}, \quad p, q \text{ integers}, \quad q \ne 0 \qquad \text{a number between } x \text{ and } y: \frac{x + y}{2}$
Adding fractions and taking a fraction of a number · Fraction sums and parts $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \qquad \frac{a}{b} \text{ of } N = \frac{a}{b} \times N$
Recurring decimals as fractions · Pure recurring decimals $0.\overline{a} = \frac{a}{9} \qquad 0.\overline{ab} = \frac{ab}{99}$

Watch out for (6)

Reading negative fractions by size
→ Rational numbers and comparing them
Thinking the numerator cannot be 0
→ Rational numbers and comparing them
Taking a fraction of the whole instead of the rest
→ Adding fractions and taking a fraction of a number
Adding the denominators
→ Adding fractions and taking a fraction of a number
Putting the whole part into the block
→ Recurring decimals as fractions
Mixing up where the bar starts
→ Recurring decimals as fractions

Irrational Numbers

Formulas (1)

Sums and products with irrational numbers · Squares and products with a root $(a + \sqrt{b})^2 = a^2 + 2a\sqrt{b} + b \qquad (a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$

Reference tables (1)

Which numbers are irrational7 rows

Number	Type	Reason
$\sqrt{2}$ , $\sqrt{3}$ , $\sqrt{5}$	Irrational	2, 3 and 5 are not perfect squares
$\sqrt{49}$	Rational	It equals 7
$\sqrt{1296}$	Rational	It equals 36
$\pi$	Irrational	Its decimal never ends and never repeats $\frac{22}{7}$ is close to $\pi$ but is a fraction, so it is rational.
$\frac{22}{7}$ , 3.14	Rational	Each is a fraction of two integers
$0.\overline{3}$	Rational	It repeats: $\frac{1}{3}$
0.101001000100001...	Irrational	It never ends and has no repeating block

A root sign alone decides nothing: check whether the number under it is a perfect square.

Watch out for (5)

Treating every root as irrational
→ Which numbers are irrational
Taking 22/7 for π
→ Which numbers are irrational
Assuming two irrationals add to an irrational
→ Sums and products with irrational numbers
Dropping the middle term when squaring
→ Sums and products with irrational numbers
Forgetting the zero case
→ Sums and products with irrational numbers

Question weightage by concept

14 concepts · 47 questions — where the marks actually sit, so you know what to drill first

Number Types and Place Value8 questions · 17%

Concept	Qs	Share
Natural numbers, whole numbers, integers, even and odd	3	6%
Place value and face value	3	6%
Counting digits and numbers made from digits	2	4%

Divisibility, Primes and Factorisation11 questions · 23%

Concept	Qs	Share
Prime factorisation	7	15%
Divisibility rules	2	4%
Prime and composite numbers	2	4%

HCF and LCM11 questions · 23%

Concept	Qs	Share
HCF and LCM from prime factors	6	13%
Euclid's division and the largest number that divides	3	6%
HCF times LCM equals the product	2	4%

Fractions and Rational Numbers10 questions · 21%

Concept	Qs	Share
Adding fractions and taking a fraction of a number	4	9%
Rational numbers and comparing them	3	6%
Recurring decimals as fractions	3	6%

Irrational Numbers7 questions · 15%

Concept	Qs	Share
Sums and products with irrational numbers	4	9%
Which numbers are irrational	3	6%

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