Number

The One Thing

Numerical reasoning requires more than obtaining a numerical answer: the method, units, accuracy, assumptions and interpretation must be appropriate to the situation. The topic combines real-number operations, estimation, percentage and financial calculations, and systematic checking of results.

Definitions and Results

• Real number: Any number that can be represented on the number line, including rational and irrational numbers. The real number system includes natural numbers, whole numbers, integers, rational numbers and irrational numbers.

• Rational number: A number that can be written as $a/b$, where $a$ and $b$ are integers and $b\ne 0$. Its decimal form terminates or repeats.

• Irrational number: A real number that cannot be written as a fraction of integers. Its decimal expansion is non-terminating and non-repeating, such as $\sqrt{2}$ or $\pi$.

• Index or exponent: A number showing how many times a base is multiplied by itself; for example,

$$3^4=3\times 3\times 3\times 3.$$ For powers with the same non-zero base, $$a^m\times a^n=a^{m+n},a^m÷a^n=a^{m-n},(a^m{)}^n=a^{mn}.$$ For non-zero $a$, $$a^0=1,a^{-n}=\frac{1}{a^n}.$$

• Scientific notation: A representation of a very large or very small number in the form

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

• Surd: An exact irrational expression involving a root that cannot be simplified to a rational number, such as $\sqrt{5}$. Exact forms such as $3\sqrt{2}$ or $\pi$ should be retained when precision is important.

• Percentage change: The change in a quantity expressed as a percentage of its original value:

$$\text{percentage change}=\frac{\text{change}}{\text{original value}}\times 100\%.$$ Percentages must be interpreted relative to a stated original or reference quantity.

• Percentage multiplier: A percentage increase by $r$, with $r$ written as a decimal, is represented by multiplying by $1+r$. A percentage decrease is represented by multiplying by $1-r$.

• Simple interest: Interest calculated only on the original principal:

$$I=Prt,$$ where $P$ is the principal, $r$ is the annual interest rate as a decimal and $t$ is time in years. The final amount is $$A=P+I.$$

• Compound interest: Interest calculated on the original amount and previously earned interest:

$$A=P{\left(1+\frac{r}{n}\right)}^{nt},$$ where $n$ is the number of compounding periods per year. For annual compounding, $$A=P(1+r{)}^t.$$

• Depreciation: A decrease in value over time, often modelled by repeated percentage reduction or a linear decrease.

• Ratio and rate: A ratio compares quantities, while a rate compares quantities with different units, such as kilometres per hour or dollars per kilogram.

• Rounding: Replacing a number with a nearby value at a specified place value; for example, $4.768$ rounded to two decimal places is $4.77$.

• Significant figures: Digits that convey the precision of a measurement, beginning with the first non-zero digit.

• Bounds: Limits describing the possible range of an exact value when a rounded or measured value is given. A value rounded to the nearest unit lies from half a unit below the rounded value up to, but not including, half a unit above it.

• Error and accuracy: Error is the difference between an approximation and an exact or accepted value. Accuracy indicates how close a result is to that value. Percentage error is

$$\frac{|\text{approximate value}-\text{exact value}|}{|\text{exact value}|}\times 100\%.$$

• Unit conversion: Changing a measurement into an equivalent unit using appropriate conversion factors while preserving the quantity.

• Financial model: A mathematical representation of a financial situation, including income, expenses, loans, savings, tax, interest and depreciation.

• Budget: A budget can be represented by

$$\text{income}-\text{expenses}.$$ A surplus is positive and a deficit is negative.

• Order of operations: Evaluate grouping symbols, then indices, then multiplication and division, and finally addition and subtraction. Within each level, work from left to right.

• Estimation: Estimation methods include compatible numbers, leading-digit estimation, rounding before calculation and using known benchmarks.

• Accuracy of answers: Answers should include appropriate units and a sensible level of rounding based on the accuracy of the given information. Exact values and approximations serve different purposes: exact forms preserve precision, while rounded values support practical measurement, comparison and communication.

Worked Methods

Simplifying powers and applying the order of operations

• Identify powers with the same non-zero base.
• For multiplication, add the exponents:

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

• For division, subtract the exponents:

$$a^m÷a^n=a^{m-n}.$$

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

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

• Apply the full order of operations: grouping symbols, indices, multiplication and division, then addition and subtraction, working from left to right within each level.
• Use $a^0=1$ and $a^{-n}=1/a^n$ only when $a\ne 0$.

Simplifying a surd

To simplify $\sqrt{72}$:

• Factor $72$ using a perfect square:

$$72=36\times 2.$$

• Separate the square root:

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

• Evaluate the perfect-square root:

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

The exact result is $6\sqrt{2}$. A decimal approximation may be used when a practical measurement or comparison is required.

Applying successive percentage changes

For two successive $10\%$ increases:

• Convert $10\%$ to the decimal $0.1$.
• Represent each increase by the multiplier $1+0.1=1.1$.
• Apply the multipliers successively:

$$1.1\times 1.1=\mathrm{1.21.}$$

• Interpret the result: the total increase is $21\%$, not $20\%$.

Successive percentage changes must therefore be applied multiplicatively rather than added directly.

Calculating simple interest

• Identify the principal $P$, annual interest rate $r$ as a decimal, and time $t$ in years.
• Substitute into

$$I=Prt.$$

• Calculate the final amount:

$$A=P+I.$$

• State the result with an appropriate currency unit and level of rounding.

Calculating compound interest

• Identify the principal $P$, annual rate $r$ as a decimal, number of compounding periods per year $n$, and time $t$ in years.
• Substitute into

$$A=P{\left(1+\frac{r}{n}\right)}^{nt}.$$

• If interest is compounded annually, use

$$A=P(1+r{)}^t.$$

• Interpret $A$ as the final amount and compare it with the original principal when determining the total interest.

For a loan repayment or investment calculation, also identify the repayment or compounding frequency and any fees. The total cost or total return may therefore differ from what the stated percentage rate alone suggests.

Modelling depreciation

• Identify the initial value and the rate or amount of decrease.
• For repeated percentage depreciation, multiply by the remaining-value multiplier $1-r$, where $r$ is the percentage decrease written as a decimal.
• Apply the multiplier for each relevant time period.
• For linear depreciation, subtract the same fixed amount for each period.
• State the final value with appropriate units and rounding.

Finding percentage error

• Identify the approximate value and the exact or accepted value.
• Find their absolute difference:

$$|\text{approximate value}-\text{exact value}|.$$

• Divide by the absolute exact value.
• Multiply by $100\%$:

$$\frac{|\text{approximate value}-\text{exact value}|}{|\text{exact value}|}\times 100\%.$$

Determining bounds after rounding

• Identify the place value used for rounding.
• Subtract half of that place-value unit from the rounded value.
• Add half of that unit to the rounded value.
• Use an inclusive lower bound and an upper bound that is not included.

For a value rounded to the nearest unit, the actual value lies from half a unit below the rounded value up to, but not including, half a unit above it.

Building and checking a financial model

• Identify the quantities involved: income, expenses, principal, rates, time, repayment or compounding frequency, fees and changing balances.
• Select the relevant model, such as a budget, simple interest, compound interest or depreciation model.
• Convert percentages to decimals and ensure that units and time periods are consistent.
• Substitute values into the appropriate formula.
• Estimate the likely size of the result before or after calculation.
• Use a calculator accurately, while retaining responsibility for choosing the correct operation and entering the expression correctly.
• Check whether the result is sensible and identify assumptions or limitations in the model.

Where It Goes Wrong

• Treating successive percentage changes as additive: two successive $10\%$ increases produce $1.1\times 1.1=1.21$, or a $21\%$ increase, rather than $20\%$.
• Applying a percentage without identifying the original or reference quantity.
• Confusing simple interest, $I=Prt$, with compound interest,

$$A=P{\left(1+\frac{r}{n}\right)}^{nt}.$$

• Omitting compounding or repayment frequency, time period or fees in a loan or investment calculation.
• Giving a rounded decimal when an exact form such as $3\sqrt{2}$ or $\pi$ is required for precision, or using inappropriate rounding for the accuracy of the given information.
• Reporting an answer without units, bounds, assumptions or a reasonableness check.

What Gets Asked

This material supports questions requiring students to:

• Classify numbers as natural, whole, integer, rational, irrational or real.
• Apply index laws, zero and negative exponents, scientific notation and the order of operations.
• Simplify surds, including $\sqrt{72}=6\sqrt{2}$, and distinguish exact forms from decimal approximations.
• Calculate percentage increases, decreases and percentage error.
• Apply successive percentage changes using multiplicative factors.
• Calculate simple interest, compound interest, final amounts, loan costs, investment returns and depreciation.
• Construct or interpret budgets, including surpluses and deficits.
• Convert units and distinguish ratios from rates.
• Round values, identify significant figures and determine bounds.
• Estimate using compatible numbers, leading digits, preliminary rounding and known benchmarks.
• Use calculators accurately while selecting suitable operations and checking whether answers are sensible.
• Evaluate financial models by considering total cost or return, compounding frequency, fees, time, changing balances, assumptions and limitations.