1 Arithmetic and numerical computation

Learning objectives

11A Decimal-form numbersRecognise and use numbers in decimal form.11B Standard-form numbersRecognise and use numbers in standard form.11C Ratios, fractions, percentages, powers and rootsUse ratios, fractions, percentages, powers and roots.11D Estimate simple calculationsEstimate results of simple calculations without a calculator.11E Trigonometric calculator functionsUse calculators to handle sin x and sin⁻¹ x, where x is in degrees.

A decimal writes a number directly by place value. Each move one place to the left makes a digit worth ten times more; each move one place to the right makes it worth ten times less. Thus $3.47=3+4/10+7/100$ , while the zeros in $0.047$ hold the tenths place empty.

First identify the place of each digit and the size the answer should have. For addition or subtraction, align decimal points so equal place values are combined. For multiplication or division, calculate the digits and then check the decimal position against an estimate. Keep the physical unit attached to the final value.

Example: a current changes from $0.38\,\mathrm{A}$ to $1.7\,\mathrm{A}$ . The increase is $1.70-0.38=1.32\,\mathrm{A}$ ; writing $1.7$ as $1.70$ makes the place-value alignment visible but does not change its value. A unit conversion is a multiplication, not a free decimal shift: $0.025\,\mathrm{A}\times1000=25\,\mathrm{mA}$ .

A leading zero prevents misreading: write $0.6$ , not $.6$ . Moving a decimal point changes the number unless a compensating scale factor is also applied. Extra trailing zeros can communicate precision, but choosing significant figures is a separate skill; here the essential check is that every digit keeps its intended place value.

In standard form, a non-zero number is written as $a\times10^n$ , where $1\leq|a|<10$ and $n$ is an integer. A positive exponent represents repeated multiplication by 10; a negative exponent represents repeated division by 10.

$45\,000=4.5\times10^4 \qquad 0.00032=3.2\times10^{-4}$

To convert into standard form, move the decimal point until the first factor lies from 1 up to but not including 10; the number of places moved gives the exponent. When multiplying, multiply the first factors and add exponents. When dividing, divide the first factors and subtract exponents. Finally normalise the first factor if it lies outside the allowed range.

Example: light travels at $3.0\times10^8\,\mathrm{m/s}$ for $3.2\times10^7\,\mathrm{s}$ . Distance $=(3.0\times3.2)\times10^{8+7}=9.6\times10^{15}\,\mathrm{m}$ , which is approximately $10^{16}\,\mathrm{m}$ . The powers combine separately from the ordinary numbers.

The exponent belongs to 10, not to the first factor. Also, $10^{-4}$ is a small positive scale factor, not a negative number. Calculator displays such as 3.2E-4 mean $3.2\times10^{-4}$ ; they do not mean $3.2\times e-4$ .

Ratios, fractions and percentages compare quantities; powers and roots describe how a quantity is combined with itself or undo that operation. The important move is to identify the relationship before calculating, then keep numerator, denominator and scale direction consistent.

Form	Meaning and operation	Compact example
ratio $a:b$	compare $a$ with $b$ ; use a scale factor	$1:4$ means multiply a diagram length by 4
fraction $a/b$	$a$ divided by $b$ ; part of a whole	$18/60=0.30$
percentage	fraction out of 100	$0.30\times100\%=30\%$
power $x^n$	multiply $x$ by itself $n$ times	$3^2=9$
root $\sqrt[n]{x}$	undo raising to power $n$	$\sqrt{49}=7$

Example: a scale says $1\,\mathrm{cm}$ on a diagram represents $4\,\mathrm{cm}$ in the laboratory. A measured diagram distance of $6.2\,\mathrm{cm}$ therefore represents $6.2\times4=24.8\,\mathrm{cm}$ . Multiplying is correct because the real distance is four times the diagram distance; reversing the ratio would shrink it.

A ratio compares quantities in a stated order, so $1:4$ is not interchangeable with $4:1$ . A percentage is dimensionless only after like quantities and compatible units are compared. Squaring and taking a square root are inverse operations, but $\sqrt{a+b}$ is not generally $\sqrt a+\sqrt b$ .

An estimate replaces awkward values with nearby, easy numbers so the calculation can be completed mentally. It should preserve the operation, sign and approximate scale of the original expression, and it should be written with $\approx$ because the rounded expression is not exactly equal to the original.

Use this sequence: round each value to one convenient significant figure → keep any powers of ten explicit → carry out the simple arithmetic → restore the unit → check whether the result has a sensible sign and order of size. Round enough to simplify, but not so much that a non-zero divisor becomes zero.

Example: one fusion reaction releases about $3\times10^{-12}\,\mathrm{J}$ . The number needed for $1\,\mathrm{J}$ is $1/(3\times10^{-12})=(1/3)\times10^{12}$ . Since $1/3\approx0.3$ , this is about $0.3\times10^{12}=3\times10^{11}$ reactions. The power of ten protects the scale during the mental calculation.

For a division, ask whether the denominator is smaller or larger than 1: dividing by a very small number should produce a very large result. An estimate is useful for predicting magnitude and catching calculator-entry errors; it is not permission to omit units or replace a requested accurate calculation with a rough answer.

The sine key maps an angle to a ratio; the inverse-sine key maps a valid ratio back to an angle. For this course, the angle $x$ is expressed in degrees, so check that the calculator shows DEG before entering either operation.

$x\;\xrightarrow{\sin}\;\sin x \qquad r\;\xrightarrow{\sin^{-1}}\;x$

To evaluate $\sin x$ , enter the angle and apply sin; for example, $\sin30^\circ=0.5$ . To recover an angle, apply sin⁻¹ to the ratio; for example, $\sin^{-1}(0.75)\approx48.6^\circ$ . On most calculators, inverse sine is accessed with SHIFT or 2nd followed by SIN.

A sine value must lie from $-1$ to $1$ , so an inverse-sine input outside that interval signals an earlier error. Check the output includes a degree interpretation and is plausible: for an acute angle, a ratio near 0 gives a small angle and a ratio near 1 gives an angle near $90^\circ$ .

$\sin^{-1}x$ means inverse sine, not the reciprocal $1/\sin x$ . RAD mode gives a numerically different angle, and a missing bracket can change what the calculator evaluates. Keep the complete ratio inside the inverse-sine operation before rounding the angle.

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1 Arithmetic and numerical computation

Learning objectives

Use decimal place value without losing the scale

Standard form keeps very large and small values manageable

Choose the numerical relationship that matches the physics

Estimate with controlled rounding, not a guess

Use sine and inverse sine in degree mode

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