Number

The One Thing

GCSE number work depends on preserving exact values while applying index laws, simplifying surds, using standard form and interpreting rounding through bounds. Correct solutions require attention to restrictions, units of accuracy, inequality signs and the distinction between exact and approximate forms.

Definitions and Results

• Natural numbers: Counting numbers such as $1,2,3$, and sometimes $0$, depending on the definition being used.
• Integer: A positive or negative whole number, or zero, such as $-4$, $0$ and $9$.
• Rational number: A number that can be written as $\frac{a}{b}$, where $a$ and $b$ are integers and $b\ne 0$. Its decimal expansion terminates or repeats.
• Irrational number: A number that cannot be written as an exact fraction. Its decimal expansion is non-terminating and non-repeating, such as $\sqrt{2}$ or $\pi$.
• Real number: Any number that can be placed on the number line, including rational and irrational numbers. Integers are rational, and rational and irrational numbers are real.
• Index or exponent: The power showing how many times a number is multiplied by itself, such as $5^3$.
• Index laws: For appropriate bases and indices,

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

• Power of a product:

$$(ab{)}^n=a^n{b}^n.$$

• Power of a quotient:

$${\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n},b\ne 0.$$

• Fractional index:

$$a^{1/n}=\sqrt[n]{a},a^{m/n}=\sqrt[n]{a^m}.$$

• Surd: An irrational root written exactly, such as $\sqrt{3}$ or $2\sqrt{5}$, rather than as a rounded decimal.
• Simplifying surds: Square factors are taken outside the root:

$$\sqrt{72}=\sqrt{36\times 2}=6\sqrt{2}.$$ Surds should remain in exact form when a decimal approximation would lose accuracy.

• Surd multiplication:

$$\sqrt{a}\times \sqrt{b}=\sqrt{ab},$$ for suitable non-negative $a$ and $b$.

• Surd division:

$$\sqrt{a}÷\sqrt{b}=\sqrt{\frac{a}{b}},$$ where $b$ is positive.

• Rationalising the denominator: Removing a surd from the denominator of a fraction.
• Simple rationalisation:

$$\frac{1}{\sqrt{a}}=\frac{\sqrt{a}}{a},a>0.$$

• Conjugate: An expression formed by changing the sign between two terms. The conjugate of $a+\sqrt{b}$ is $a-\sqrt{b}$. Multiplying by the conjugate can remove a surd from a denominator.
• Standard form: A number written as

$$A\times 10^n,$$ where $1\le A<10$ and $n$ is an integer.

• Significant figures: Digits contributing to the precision of a number, beginning with the first non-zero digit.
• Decimal places: The number of digits shown after the decimal point.
• Rounding: Replacing a number with a nearby value at a specified accuracy. A following digit of $5$ or more rounds up; a digit below $5$ leaves the previous digit unchanged.
• Truncation: Cutting off digits after a chosen place without rounding.
• Upper bound: The greatest possible value of a rounded measurement. For rounding to the nearest unit:

$$\text{upper bound}=\text{stated value}+\mathrm{0.5.}$$

• Lower bound: The smallest possible value of a rounded measurement. For rounding to the nearest unit:

$$\text{lower bound}=\text{stated value}-\mathrm{0.5.}$$

• Interval notation for bounds: If $x$ is rounded to the nearest unit,

$$\text{lower bound}\le x<\text{upper bound}.$$ The lower bound is included, but the upper bound is not.

• Error interval: The range of actual values that could have produced a stated rounded or truncated value.

Worked Methods

1. Classifying numbers

• Identify whether the number is a counting number, integer, rational or irrational.
• Use the containment relationships:

- integers are rational; - rational and irrational numbers are real.

• For rationality, determine whether the number can be expressed as $\frac{a}{b}$, with $a,b$ integers and $b\ne 0$.
• A terminating or repeating decimal is rational; a non-terminating, non-repeating decimal is irrational.

2. Applying index laws

• Check that the powers have the same non-zero base where required.
• For multiplication, add the indices:

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

• For division, subtract the indices:

$$a^m÷a^n=a^{m-n}.$$

• For a power of a power, multiply the indices:

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

• For products and quotients, apply the power to each factor:

$$(ab{)}^n=a^n{b}^n,{\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}.$$

• Apply special cases:

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

• For fractional indices, convert to roots:

$$a^{1/n}=\sqrt[n]{a},a^{m/n}=\sqrt[n]{a^m}.$$

3. Simplifying and manipulating surds

• Factor the number under the root to identify the largest square factor.
• Take the square root of that factor outside the surd.
• For example:

$$\sqrt{72}=\sqrt{36\times 2}=6\sqrt{2}.$$

• Multiply surds using:

$$\sqrt{a}\times \sqrt{b}=\sqrt{ab}.$$

• Divide surds using:

$$\sqrt{a}÷\sqrt{b}=\sqrt{\frac{a}{b}},$$ where $b>0$.

• Retain the exact surd form rather than using a rounded decimal when accuracy is required.

4. Rationalising a denominator

For a simple denominator:

• Multiply the numerator and denominator by the same surd:

$$\frac{1}{\sqrt{a}}\times \frac{\sqrt{a}}{\sqrt{a}}=\frac{\sqrt{a}}{a},a>0.$$

• For a denominator such as $a+\sqrt{b}$, multiply the numerator and denominator by its conjugate $a-\sqrt{b}$:

$$\frac{c}{a+\sqrt{b}}\times \frac{a-\sqrt{b}}{a-\sqrt{b}}.$$

• Use the product of conjugates to remove the surd from the denominator.

5. Converting to standard form

• Move the decimal point until the first number is at least $1$ but less than $10$.
• Count the number of places moved.
• Moving the decimal point left gives a positive power of $10$.
• Moving it right gives a negative power of $10$.
• Write the result as $A\times 10^n$, with $1\le A<10$ and integer $n$.

For calculations:

• Multiplication:

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

• Division:

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

• Addition or subtraction: first rewrite both numbers using the same power of $10$, then add or subtract the coefficients.

6. Rounding and truncation

• Identify the required accuracy: significant figures, decimal places or a stated unit.
• For rounding, inspect the next digit:

- $5$ or more: increase the preceding digit; - below $5$: leave the preceding digit unchanged.

• For truncation, remove all later digits without changing the final retained digit.
• For a number rounded to the nearest $0.1$, the actual value lies within $0.05$ of the stated value.
• For a number rounded to $d$ decimal places, the half-unit of accuracy is

$$\frac{1}{2}\times 10^{-d}.$$

7. Finding bounds and error intervals

• Identify the unit to which the value was rounded.
• Subtract half that unit to obtain the lower bound.
• Add half that unit to obtain the upper bound.
• Write the interval with the lower bound included and the upper bound excluded.

Examples:

• A number stated as $10$, rounded to the nearest $10$, satisfies

$$5\le x<15.$$ Here, $$\text{lower bound}=10-5=5,\text{upper bound}=10+5=15.$$

• For rounding to the nearest unit:

$$\text{lower bound}=\text{stated value}-0.5,$$ $$\text{upper bound}=\text{stated value}+\mathrm{0.5.}$$

• For rounding to the nearest $10$:

$$\text{lower bound}=\text{stated value}-5,$$ $$\text{upper bound}=\text{stated value}+5.$$

8. Using bounds in calculations

• Find the lower and upper bounds of each quantity.
• Consider the operation being performed.
• For positive quantities:

- a maximum product usually uses upper bounds; - a minimum product usually uses lower bounds.

• For a quotient of positive quantities:

- the maximum usually uses the greatest numerator divided by the least denominator; - the minimum usually uses the least numerator divided by the greatest denominator.

• Keep full calculator accuracy during intermediate steps.
• Round only at the final stage unless instructed otherwise.
• Check the size and form of the answer to ensure that rounding has not introduced an avoidable error.

Where It Goes Wrong

• Applying index laws to different bases, rather than checking that the bases are handled consistently.
• Forgetting the restrictions $a\ne 0$ in $a^0=1$, $b\ne 0$ in ${\left(\frac{a}{b}\right)}^n$, or $b>0$ in surd division.
• Replacing an exact surd with a rounded decimal when the question requires an exact value.
• Leaving a surd in the denominator instead of rationalising it, or using the wrong conjugate for a denominator such as $a+\sqrt{b}$.
• Writing a standard-form coefficient outside the range $1\le A<10$, or using the wrong sign for the power of $10$.
• Treating an upper bound as included: the correct interval is lower bound $\le x<$ upper bound.

What Gets Asked

This material supports questions requiring students to:

• classify examples as natural numbers, integers, rational numbers, irrational numbers or real numbers;
• apply index laws, including zero, negative and fractional indices;
• convert fractional indices into roots;
• simplify, multiply and divide surds;
• rationalise denominators using a simple surd or a conjugate;
• convert numbers to and from standard form;
• multiply, divide, add and subtract numbers in standard form;
• round to specified significant figures or decimal places;
• distinguish rounding from truncation;
• find lower bounds, upper bounds and error intervals;
• interpret intervals such as $5\le x<15$;
• calculate maximum and minimum values using bounds;
• retain full calculator accuracy until the final stage;
• check whether an exact answer, a standard-form answer or an appropriately rounded answer is required.