Numbers, Powers and Logarithms — IMAT Mathematics Formulas and Traps

Number Sets, Primes and Ordering

Prime factorisation, highest common factor and lowest common multiple

HCF and LCM of two numbers

\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b

$a, b$ two positive whole numbers

Fractions, recurring decimals and putting numbers in order

Order for 0 < x < 1

0 < x^2 < x < \sqrt{x} < 1 < \frac{1}{\sqrt{x}} < \frac{1}{x}

The number sets ℕ, ℤ, ℚ and ℝ

Set	Symbol	What it contains	Examples
Natural numbers	$\mathbb{N}$	The counting numbers (Italian books usually include 0)	0, 1, 2, 3, ...
Integers	$\mathbb{Z}$	Whole numbers: positive, negative and zero	$-4,\ 0,\ 7$
Rational numbers	$\mathbb{Q}$	Fractions of integers; decimals that end or repeat	$\frac{3}{8} = 0.375,\ 0.\overline{3} = \frac{1}{3},\ -5$
Irrational numbers	$\mathbb{R} \setminus \mathbb{Q}$	Decimals that never end and never repeat	$\sqrt{2},\ \pi,\ e,\ \sqrt{12}$
Real numbers	$\mathbb{R}$	Rationals and irrationals together: every point of the number line	All of the above

Each row's numbers also belong to every larger set: 7 is natural, integer, rational and real.

Common traps

22/7 is not π

\frac{22}{7}

is a fraction of two integers, so it is rational. It is only an approximation of

\pi

, which is irrational. An option offering

\frac{22}{7}

as the irrational number is wrong.

HCF uses the lowest powers, LCM the highest

Swapping the rule turns the HCF into the LCM, and IMAT puts that number among the options. The HCF can never be larger than the smallest of the numbers; the LCM can never be smaller than the largest.

Squaring does not always make a number bigger

For a number between 0 and 1, the square is smaller and the square root is larger:

0.5^2 = 0.25

but

\sqrt{0.25} = 0.5

. The intuition that powers grow is only true above 1.

Powers, Fractional Exponents and Surds

Learn this subtopic in the notes

Laws of powers and rewriting to a common base

Laws of powers

a^m a^n = a^{m+n} \qquad \frac{a^m}{a^n} = a^{m-n} \qquad (a^m)^n = a^{mn}

$a$ the base, the same in every factor
$m, n$ the exponents

Fractional exponents and nth roots

Fractional exponent

a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}

$n$ the root (denominator of the exponent)
$m$ the power (numerator of the exponent)

Simplifying surds and rationalising the denominator

Rationalising with the conjugate

\frac{a}{b+\sqrt{c}} = \frac{a\,(b-\sqrt{c})}{b^2 - c}

$b - \sqrt{c}$ the conjugate of $b + \sqrt{c}$

Common traps

Adding equal powers does not add the exponents

2^{10} + 2^{10} = 2 \times 2^{10} = 2^{11}

, not

2^{20}

and not

4^{10}

. Exponents add only when powers of the same base are multiplied.

A negative exponent does not make the number negative

4^{-1/2} = \frac{1}{\sqrt{4}} = \frac{1}{2}

, a positive number. The minus sign in an exponent means take the reciprocal.

The root of a sum is not the sum of the roots

\sqrt{9 + 16} = \sqrt{25} = 5

, not

3 + 4 = 7

. Roots split over a product or a quotient, never over a sum or a difference.

The conjugate denominator is b² minus c

(b + \sqrt{c})(b - \sqrt{c}) = b^2 - c

. Writing

b^2 + c

is the usual slip, and an option built on it is always present.

Scientific Notation and Logarithms

Learn this subtopic in the notes

Scientific notation and orders of magnitude

Scientific notation

x = a \times 10^{n}, \qquad 1 \le a < 10, \quad n \in \mathbb{Z}

$a$ the digit part (mantissa)
$n$ the power of ten, a whole number

Logarithm: definition, log₁₀ and ln

Definition of the logarithm

\log_b x = y \quad\text{means}\quad b^{y} = x

$b$ the base, positive and not 1
$x$ the argument, positive

Laws of logarithms and change of base

Laws of logarithms

\log(xy) = \log x + \log y \qquad \log\frac{x}{y} = \log x - \log y \qquad \log x^{k} = k\log x \qquad \log_b x = \frac{\log x}{\log b}

Solving exponential equations and rearranging log relations

Exponential equation by logs

a^{x} = c \;\Rightarrow\; x = \frac{\log c}{\log a} \qquad (c > 0)

Common traps

15 × 10⁴ is not in scientific notation

The digit part must be at least 1 and less than 10.

15 \times 10^4 = 1.5 \times 10^5

: moving the decimal point one place left raises the exponent by one. Forgetting this gives an option that is ten times too small or too large.

A negative logarithm is fine; a negative argument is not

\log 0.01 = -2

is perfectly good: the number is between 0 and 1. But

\log(-2)

and

\log 0

do not exist, because no power of a positive base is zero or negative.

The log of a sum does not split

\log(x + y)

is not

\log x + \log y

; that sum equals

\log(xy)

. Likewise

\frac{\log x}{\log y}

is not

\log \frac{x}{y}

: the first is a change of base, the second a difference of logs.

The constant in a log relation becomes a factor

\log y = 2\log x + 1

gives

y = 10x^2

, not

y = x^2 + 1

. The 1 is

\log 10

, and logs that add come from numbers that multiply.

Absolute Value, Percentages and Ratios

Learn this subtopic in the notes

Absolute value as distance from zero

Absolute value

|x| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}

Percentages: change, reverse percentage and successive changes

Percentage change as a multiplier

\text{new} = \text{original} \times \left(1 \pm \frac{p}{100}\right)

$p$ the percentage rise (+) or fall (minus)

Ratios and sharing in a given ratio

Sharing in the ratio a : b

\text{parts} = \frac{a}{a+b}\,T \quad\text{and}\quad \frac{b}{a+b}\,T

$T$ the total being shared

Common traps

Minus x is not always negative

|x| = -x

is true for every

x \le 0

: if

x = -5

, then

-x = 5

. An option claiming

|x| = -x

is impossible is wrong.

A rise and a fall of the same percentage do not cancel

Up 20% then down 20% gives

1.2 \times 0.8 = 0.96

: a 4% loss overall. The fall is a percentage of the larger, already-raised value.

Adding the same amount to both parts changes the ratio

Adding 1 to each part of

2 : 3

gives

3 : 4

, a different ratio. Only multiplying or dividing both parts by the same number keeps a ratio.

Numbers, Powers and Logarithms formulas

Number Sets, Primes and Ordering

Prime factorisation, highest common factor and lowest common multiple

Fractions, recurring decimals and putting numbers in order

The number sets ℕ, ℤ, ℚ and ℝ

Common traps

Powers, Fractional Exponents and Surds

Laws of powers and rewriting to a common base

Fractional exponents and nth roots

Simplifying surds and rationalising the denominator

Common traps

Scientific Notation and Logarithms

Scientific notation and orders of magnitude

Logarithm: definition, log₁₀ and ln

Laws of logarithms and change of base

Solving exponential equations and rearranging log relations

Common traps

Absolute Value, Percentages and Ratios

Absolute value as distance from zero

Percentages: change, reverse percentage and successive changes

Ratios and sharing in a given ratio

Common traps

More IMAT Mathematics formula sheets