Number

The One Thing

Number work requires selecting an appropriate representation and applying the operation suited to its structure. Exact forms preserve accuracy, while percentages, standard form, approximation and bounds support efficient comparison and practical calculation.

Definitions and Results

• Natural numbers: The positive counting numbers, usually $1,2,3,\dots$; some conventions also include $0$.
• Integers: Positive and negative whole numbers, including zero.
• Rational numbers: Numbers that can be written as $\frac{p}{q}$, where $p$ and $q$ are integers and $q\ne 0$. Their decimal expansions terminate or recur.
• Irrational numbers: Numbers that cannot be written as a fraction of two integers, such as $\sqrt{2}$ and $\pi$. Their decimal expansions are non-terminating and non-recurring.
• Real numbers: All rational and irrational numbers represented on the number line. Natural numbers, integers, rational numbers and irrational numbers are all contained within the real numbers.
• Prime number: A whole number greater than $1$ with exactly two positive factors: $1$ and itself.
• Factor: A number that divides another number exactly, leaving no remainder.
• Multiple: A number obtained by multiplying a given number by an integer.
• Highest common factor: The greatest factor shared by two or more numbers.
• Lowest common multiple: The smallest positive multiple shared by two or more numbers.
• Prime factorisation: Expressing a number as a product of prime numbers. For positive integers, the HCF uses common prime factors with the smallest powers, while the LCM uses all prime factors with the greatest powers.
• Set: A well-defined collection of objects, numbers or elements.
• Element: An object or number belonging to a set, written using $\in$.
• Universal set: The complete set of objects being considered, usually written as $\xi$ or $U$.
• Empty set: A set containing no elements, written as $\varnothing$ or $\{\}$.
• Union: The set of elements in $A$ or $B$ or both, written $A\cup B$.
• Intersection: The set of elements common to both $A$ and $B$, written $A\cap B$.
• Complement: The elements in the universal set that are not in $A$, written $A^{\prime }$ or $A^c$.
• Subset: A set whose every element is also in another set, written $A\subseteq B$.
• Set-builder notation: A description of a set using a condition, for example $\{x\in \mathbb{Z}:x>2\}$.
• Index or exponent: The small raised number showing how many times a base is multiplied by itself.
• Index laws: For a common non-zero base,

$$a^m{a}^n=a^{m+n},\frac{a^m}{a^n}=a^{m-n},(a^m{)}^n=a^{mn},$$ $$a^0=1(a\ne 0),a^{-n}=\frac{1}{a^n}.$$

• Fraction: A number written as a numerator divided by a non-zero denominator.
• Proper fraction: A fraction whose numerator is smaller than its denominator.
• Improper fraction: A fraction whose numerator is greater than or equal to its denominator.
• Equivalent fractions: Fractions with different appearances but the same value, formed by multiplying or dividing the numerator and denominator by the same non-zero number.
• Percentage: A proportion expressed out of $100$; $x\%=\frac{x}{100}$.
• Percentage change:

$$\frac{\text{new value}-\text{original value}}{\text{original value}}\times 100\%.$$

• Multiplier: A factor used to increase or decrease a value. An increase of $r\%$ uses $1+\frac{r}{100}$; a decrease of $r\%$ uses $1-\frac{r}{100}$.
• Square root: A number that gives the original number when multiplied by itself. $\sqrt{a}$ denotes the principal non-negative square root of $a$.
• Surd: An irrational root written exactly, such as $\sqrt{3}$, rather than as a decimal approximation.
• Standard form: A number written as $A\times 10^n$, where $1\le A<10$ and $n$ is an integer.
• Rounding: Replacing a number with a nearby value to a stated degree of accuracy.
• Significant figures: Digits carrying meaningful information, beginning with the first non-zero digit.
• Decimal places: The number of digits retained after the decimal point.
• Bounds: Limits showing the range of possible actual values for a rounded number.
• Error interval: The interval containing all possible original values after rounding.
• Set cardinality rule: For finite sets,

$$n(A\cup B)=n(A)+n(B)-n(A\cap B).$$

• Fraction operations: Provided denominators are non-zero,

$$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd},$$ $$\frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd},$$ $$\frac{a}{b}÷\frac{c}{d}=\frac{ad}{bc}.$$

• Standard-form operations:

$$(A\times 10^m)(B\times 10^n)=AB\times 10^{m+n},$$ $$\frac{A\times 10^m}{B\times 10^n}=\frac{A}{B}\times 10^{m-n}.$$ For addition and subtraction, rewrite the numbers with the same power of $10$.

• Bounds for rounding: If a quantity is rounded to the nearest unit $u$, its actual value lies within half of $u$ on either side of the rounded value:

$$\text{lower bound}\le \text{actual value}<\text{upper bound}.$$ For rounding to the nearest $10$, the lower bound is $5$ below and the upper bound is $5$ above the stated value; the upper bound is excluded.

Worked Methods

Classifying numbers

• Determine whether the number is a counting number, whole number, fraction of integers, or a non-fractional real number.
• Classify it in the most specific applicable set.
• Remember that a number may belong to several nested sets: natural numbers are integers, integers are rational, and rational and irrational numbers are real.

For example, $\sqrt{2}$ is irrational because it cannot be written as a fraction of two integers. It is therefore real but not rational.

Prime factorisation, HCF and LCM

• Express each positive integer as a product of prime factors.
• For the HCF, select only the prime factors common to all numbers, using the smallest powers.
• For the LCM, select every prime factor appearing, using the greatest powers.
• Multiply the selected factors.

Set notation and Venn diagrams

• Identify the universal set, written as $\xi$ or $U$.
• Place elements in the appropriate sets.
• Use overlapping circles for sets and their intersections.
• Use $A\cup B$ for elements in either set or both.
• Use $A\cap B$ for elements common to both sets.
• Use $A^{\prime }$ or $A^c$ for elements in the universal set outside $A$.
• Use $A\subseteq B$ when every element of $A$ is also in $B$.
• For counting, apply

$$n(A\cup B)=n(A)+n(B)-n(A\cap B),$$ subtracting the intersection to avoid double-counting.

A set may also be described by set-builder notation, for example: $$\{x\in \mathbb{Z}:x>2\}.$$

Applying index laws

• Check that the bases are the same and non-zero where required.
• When multiplying powers, add the indices:

$$a^m{a}^n=a^{m+n}.$$

• When dividing powers, subtract the indices:

$$\frac{a^m}{a^n}=a^{m-n}.$$

• For a power raised to a power, multiply the indices:

$$(a^m{)}^n=a^{mn}.$$

• Use

$$a^0=1(a\ne 0)$$ and $$a^{-n}=\frac{1}{a^n}.$$

Manipulating fractions

• For addition or subtraction, find a common denominator.
• Multiply each numerator by the factor used to produce the common denominator.
• Add or subtract the numerators and retain the common denominator:

$$\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}.$$

• For multiplication, multiply numerators and denominators:

$$\frac{a}{b}\times \frac{c}{d}=\frac{ac}{bd}.$$

• For division, multiply by the reciprocal of the second fraction:

$$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\times \frac{d}{c}=\frac{ad}{bc}.$$

• Simplify the resulting fraction where possible.

Converting fractions and percentages

• To convert a fraction to a percentage, multiply by $100\%$.
• To convert a percentage to a fraction, write it over $100$ and simplify.
• Treat percentage change as a comparison with the original value:

$$\frac{\text{new value}-\text{original value}}{\text{original value}}\times 100\%.$$

For repeated percentage changes:

• Convert each change into a multiplier.
• Multiply the successive multipliers.
• Apply the combined multiplier to the original value.
• Do not simply add the percentages.

Using square roots and surds

• Identify whether an exact form is required.
• Retain a root such as $\sqrt{3}$ as a surd when a decimal approximation would lose exactness.
• Use a calculator approximation only when the question permits or requires a decimal answer.

Using standard form

• Write each number as $A\times 10^n$, with $1\le A<10$.
• For multiplication, multiply the coefficients and add the powers:

$$(A\times 10^m)(B\times 10^n)=AB\times 10^{m+n}.$$

• For division, divide the coefficients and subtract the powers:

$$\frac{A\times 10^m}{B\times 10^n}=\frac{A}{B}\times 10^{m-n}.$$

• For addition or subtraction, first rewrite the terms with the same power of $10$.
• Re-adjust the coefficient if it is not within $1\le A<10$.

Rounding and significant figures

• Identify the required degree of accuracy: nearest integer, a stated number of significant figures, or a stated number of decimal places.
• Locate the digit to retain.
• Inspect the following digit.
• Round up if that digit requires it; otherwise retain the digit.
• State the degree of accuracy clearly.

Finding error intervals and bounds

• Identify the rounding unit.
• Take half of that unit.
• Subtract it from the rounded value to obtain the lower bound.
• Add it to obtain the upper bound.
• Exclude the upper bound when the rounded value itself is assigned to the lower interval endpoint.

For a value rounded to the nearest $10$, the possible actual values are $5$ below up to, but not including, $5$ above the stated value.

For products and quotients:

• Determine whether the quantities are positive or negative.
• Test the possible endpoint combinations.
• Select the smallest and largest resulting values.
• Express the result with the appropriate inequalities.

Where It Goes Wrong

• Treating an irrational number such as $\sqrt{2}$ or $\pi$ as rational; irrational decimal expansions are non-terminating and non-recurring.
• Applying index laws when the bases are not the same non-zero base.
• Adding or subtracting fractions without first using a common denominator, or dividing by a fraction without multiplying by its reciprocal.
• Calculating repeated percentage changes by adding the percentages instead of using successive multipliers.
• Adding or subtracting standard-form numbers without first rewriting them with the same power of $10$.
• Giving a decimal approximation when an exact fraction, surd or standard-form expression is required, or omitting the stated degree of accuracy.

What Gets Asked

This material supports questions requiring students to:

• Classify numbers as natural, integer, rational, irrational, real or prime.
• Find factors, multiples, prime factorisations, HCFs and LCMs.
• Interpret and complete set notation, set-builder notation and Venn diagrams.
• Calculate unions, intersections, complements, subsets and quantities using

$$n(A\cup B)=n(A)+n(B)-n(A\cap B).$$

• Simplify powers using the index laws, including zero and negative indices.
• Add, subtract, multiply and divide fractions.
• Convert between fractions, decimals and percentages.
• Calculate percentage change and repeated percentage changes using multipliers.
• Use square roots and retain exact surd forms such as $\sqrt{3}$.
• Multiply, divide, add and subtract numbers in standard form.
• Round to a specified number of significant figures or decimal places.
• Construct error intervals and determine upper and lower bounds for calculations involving rounded values.
• Check the size, sign, units and degree of accuracy of a final answer.