Arithmetic and Numerical Computation (M0)
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You are expected to be familiar with the units used for various quantities in A-level Physics.
Ensure that you are familiar with all the following quantities:
You may also see some of these units combined with prefixes to handle very large and very small numbers.
For example, the milligram (mg) is a unit for very small masses;
These units may be combined into more complex units.
For example, specific heat capacity might be measured in joules per kilogram per Kelvin
Temperature can be measured in several different units. In Science, the two important units of temperature are:
- degrees Celsius where water freezes at and boils at
- Kelvin where is the minimum possible temperature, known as absolute zero, and an increase of is equal to an increase of
Since absolute zero is at approximately
temperature in Kelvin = temperature in degrees Celsius + 273
Degrees Celsius is the most common unit in everyday life and in Biology.
Kelvin is more commonly used in Physics, particularly in formulas where a quantity is proportional to temperature.
To make it easier to think about and discuss very large and very small numbers, scientists use prefixes before the units.
For example, a transistor in your phone might be metres wide. This can be written as 10 nano metres, where nano is a prefix meaning ‘one billionth’.
You need to recognise and use the following prefixes (and their abbreviations) in your exams.
For example:
- Four million joules is four megajoules, written
- One millionth of a gram is a microgram, written
- 0.05 seconds can be expressed in milliseconds as
When converting between units of length, you need to multiply or divide by a power of ten.
An example of this can be analysed below.
For the related conversion between units of area, you need to apply this conversion twice, because area is a two-dimensional quantity in two dimensions.
For the related conversion between units of volume, you need to apply this conversion three times, because volume is a three-dimensional quantity
Question walkthrough
Convert between units of area and units of volume.
Converts a land area from km² to m² and to cm² in standard form, then converts a volume from mL to m³, deriving the correct power-of-ten conversion factor for area and volume units.
One litre is a volume equal to a cube with sides of This is a cubic decimetre, written It is also equal to
There are 1000 litres in
The standard prefixes can be used for litres. For example, a raindrop might be which could be written
Remember that when converting from a larger unit to a smaller unit, such as from kilograms to grams, you need more of the smaller unit, so you must have more of the smaller unit.
Conversely, converting from a smaller unit to a larger unit will result in a lesser numerical value of the measurement, even though the actual quantity remains the same.
Quantities in a formula must have consistent units of measurement.
For example, if one length, area, volume, speed, or other quantity uses kilometres, the other quantities must also use kilometres rather than metres or centimetres.
Consider the formula for speed (rearranged to make time the subject):
How long does it take an athlete to run 2.4 kilometres at 4 metres per second?
Many quantities have units with a negative exponent, such as the following:
You can read these units as the first quantity per the second quantity, as written in the ‘In Words’ column.
It is useful to know that these units can also be written using a slash (‘/’), as written in the ‘Alternative’ column.
A rate quantifies how much something changes in a fixed time interval, such as one second. Rates always need units with a negative exponent.
For example, the rate that people enter a stadium before a football match could be measured in people per minute, abbreviated as or as people/min.
Question walkthrough
Determine the correct units for rates.
Converts a mass from tonnes to kilograms, then divides by time to find a rocket’s fuel consumption rate in kg per second, deriving the correct units along the way.
The number can be written in many ways, including:
While all of these expressions are equal to one of them is unique in that the value on the left (in bold) is between one and ten. This is known as its standard form.
A number is written in standard form as:
Where:
- is any integer, and
- shows the significant figures.
Any number (except zero) can be written in standard form. This is especially useful for very large and very small numbers, as it avoids writing too many zeroes.
It is important to know how to read and write numbers in standard form.
When numbers are written in their standard form, it is easy to compare their sizes:
- The number with the larger exponent, is always the largest number
- If two numbers have the same exponent, you can compare the significant digits in
For example, consider the following inequality:
The number would be much smaller than all of these.
The number would be much larger than all of these.
When working with standard form and multiplication, you will sometimes need the following power law:
It is useful to know that means the equation is true for all values of the variables. This particular statement is true for any values of and
To see why this is true, suppose that and means the multiplication of 3 copies of 10. means the multiplication of 4 (more) copies. This is a total of 7 copies of 10.
This is true for any values of and
It is useful to know that this is also true when 10 is replaced by any other value, for example, with
When working with standard form and division, you will sometimes need the following power law:
It is useful to know that means the equation is true for all values of the variables. This particular statement is true for any values of and
To see why this is true, suppose that and means the multiplication of 7 copies of 10. means the multiplication of 2 (more) copies. Those 2 copies cancel out 2 of the 7 original copies, leaving only 5 copies.
This is true for any values of and .
It is useful to know that this is also true when 10 is replaced by any other value, for example, with :
When working with standard form and exponents, you will sometimes need the following power law:
It is useful to know that means the equation is true for all values of the variables. This particular statement is true for any values of and
To see why this is true, suppose that and means the multiplication of 4 copies of 10. Raising this expression to the power of 3 means these 4 10s are repeated a total of 3 times, for a total of 12 copies.
This is true for any values of and
It is useful to know that this is also true when 10 is replaced by any other value, for example with
When raising a number in standard form to any power, remember that the significant digits and the power of ten must both be raised to this power:
In your exam, you may write your answer either in standard form or as an ordinary number, unless the question requests a specific way to write the answer.
For example, the number may be written like that, or in standard form as
Question walkthrough
Perform a calculation in standard form
Squares a side length given in standard form, then adjusts the coefficient and power of ten to write the resulting area in proper standard form.
Question walkthrough
Perform a calculation in standard form II
Substitutes values in standard form into R=st/u, then converts the result into standard form by adjusting the coefficient and power of ten.
Numbers are often presented to a certain number of significant figures, especially if the number is derived from a measurement, to indicate the precision of that number.
For example, and each have three significant digits, and are more precise than which has only two.
When converting between standard form and ordinary numbers, you must retain the same precision by including the same amount of significant figures, even if the last digit is a zero.
Units such as metres and grams can include prefixes to make them much larger or smaller.
For example, millimetre and kilogram.
The base SI units are these units without prefixes, with one exception: the base SI unit for mass is the kilogram (not the gram).
When writing a number in standard form, use the base SI unit. This is because standard form and prefixes are different tools for handling small and large numbers, and it is simplest if they are not mixed.
In standard form, small numbers such as have a negative power of ten.
When dividing by such a number, the relevant power law tells you to subtract this negative number. Ensure you do this correctly, as it is a common mistake among students.
Numbers can be rounded to a specified amount of decimal places or significant figures. Either of these specifies the precision of the number.
For example, 0.0077 is rounded to 4 decimal places, or to 2 significant figures.
Calculations with rounded numbers should be rounded to a suitable precision:
- Round additions and subtractions to the number of decimal places of the least precise value.
- Round multiplications and divisions to the number of significant figures of the least precise value.
An example of this is which should be rounded to 0.129 (3 d.p.) so it doesn’t have more decimal places than 0.121.
Another example of this is which should be rounded to 0.00093 (2 s.f.) so it doesn’t have more significant figures than 0.0077.
Your scientific calculator may be able to convert numbers to and from standard form.
You should familiarise yourself with this functionality, so you can use it easily in your exams.
For example, on a Casio calculator, press SHIFT + SET UP and find the setting to change to Sci mode, which stands for ‘scientific notation’ and displays numbers in standard form.
Ask your friends or teacher, or check online for instructions, if you need help checking what mode your calculator is in, or how to change modes.
Try typing 0.0088 on your calculator, and then follow the instructions to convert it to standard form. The result should be
Question walkthrough
Round numbers to an appropriate amount of decimal places or significant figures.
Adds a smartphone’s mass to its packaging mass, determines the appropriate precision from the least precise measurement, then multiplies by the exact quantity of 90 and rounds only the final answer.
Fractions are equivalent if they represent the same quantity.
For example, and are equivalent to each other.
You can convert any fraction to another, equivalent fraction, by multiplying or dividing both the numerator and denominator by the same number.
This is useful when simplifying fractions to an equivalent fraction that uses smaller numbers.
For example:
If two fractions have the same denominator, you can add or subtract them by adding or subtracting their numerators.
For example,
If you need to add or subtract fractions with different denominators, you must first convert them into equivalent fractions with the same denominator.
For example:
Ratios are equivalent if they represent the same proportion.
For example, and are equivalent to each other.
You can convert any ratio to another, equivalent fraction, by multiplying or dividing each part by the same number.
This is useful when simplifying ratios to an equivalent ratio that uses smaller numbers.
For example:
Question walkthrough
Create ratios from various information
Converts mixed time units into a simplified ratio in part (a), then builds a ratio from relative statements (‘twice as long as’, ‘half as long as’) using an arbitrary starting value in part (b).
You can use a ratio to divide a quantity into parts.
For example, to divide into three parts in the ratio
To do this:
1) Determine the number of parts:
2) Use this to find the size of one part by dividing the total amount by the number of parts:
3) Multiply each part of the ratio by this size:
Ratios are often presented in the form for some value of
In these cases, be aware that there are parts in total.
The 1 represents of the whole, not
Question walkthrough
Use ratios to divide quantities
Uses the ratio 3:5:2 to find the red paint needed given a known blue paint volume, then finds the blue paint needed for a 3-litre batch of the mixture.
Numbers can be expressed in various forms, each suited for different contexts and calculations. Understanding these representations is crucial for effective mathematical communication and problem-solving.
For example, all of the following describe the same number:
You need to understand numbers written in any of these ways, and how to convert numbers to decimals, percentages, and standard form.
The way you write the number may depend on:
- what form related numbers were provided to you in,
- what you are trying to do with these numbers’ and
- how the question might ask you to present your answer.
The simplest way to solve most calculations involving percentage change is to convert the percentage to a multiplier.
You can find the multiplier by applying the percentage change to the number 1.
For example:
- An increase of has a multiplier of 1.50.
- A decrease of has a multiplier of 0.50.
To increase 2000 by multiply
If an unknown value has been increased by to 3000, find the old, unknown value by dividing by the multiplier:
Be careful when finding the old value before a percentage change.
For example, suppose that in the past month, the number of bees in a hive increased by and is now 440 bees. What was the previous value?
Although the old value is clearly smaller than 440, you must use the multiplier for a percentage increase (1.1) and divide by that to find the old value.
Question walkthrough
Calculate quantities using percentage change.
Uses percentage multipliers to find one farmer’s harvest from another’s known percentage increase and a third’s from a percentage decrease, then sums all three totals.
A percentage change tells you the amount to add or subtract from the original amount. It is different to the percentage of a number.
For example, suppose an electrical vehicle can travel for 200 miles on a single charge, but a newer model can travel further. How far can the new model travel?
Many diagrams illustrate things that are much larger or smaller than the paper or screen. These are scale drawings, and they use a ratio to describe how distances on the diagram relate to distances in reality.
For example:
- Maps
- Photos of tissues and microorganisms (in Biology)
- Diagrams of large or small machines
According to the scale on this map, a distance of on the map represents on the ground.
Assess if your calculated answer for any quantity, whether in scientific contexts or daily life, is reasonable by applying your scientific understanding or general knowledge.
For example, suppose you are estimating how much petrol a car will need for its journey to a beach about 80 miles away. You know that a typical car travels about 40 miles per litre of fuel. However, you incorrectly multiply these quantities, rather than divide them:
When you don’t need a precise result from a calculation, you can do a quick estimate. This is useful for checking whether your answer is reasonable and faster than using a calculator.
There are many good ways to estimate.
You can round each value to one significant figure or to another number that you consider convenient.
For example:
The exact answer is confirming that the estimate was quite accurate.
When estimating, you should round each value to a number that is both:
- close (so the estimate will be accurate enough)
- easy for arithmetic (so you can easily complete the calculation)
When estimating a multiplication, determine if the estimate is likely to be an overestimate or an underestimate:
- Rounding either value up causes an overestimate.
- Rounding either value down causes an underestimate, for the same reason.
For example, estimating gives an overestimate, because the correct answer is
When estimating a division, the opposite is true for the denominator: rounding up results in an underestimate, and rounding down causes an overestimate.
For example, estimating gives an overestimate, because the correct answer is
It is important to note that most estimation requires rounding multiple values. Depending on which values are rounded up or down, the result could be clearly an overestimate, clearly an underestimate, or not obvious.
Question walkthrough
Estimate a result, and determine whether it’s an underestimate or overestimate
,
You can find the values of sin, cos, or tan using your scientific calculator, by pressing the buttons with those names.
However, these functions can be used in both degrees and in radians. Your calculator will have a setting to change between degrees mode and radian mode. You must use the correct mode. Otherwise, your answers will be incorrect!
For example, on a Casio calculator, press + and find the setting to change the angle unit. A small ‘D’ or ‘R’ will persist on the display so you can immediately check the mode.
If you need help checking the mode your calculator is in or learning how to change modes, refer to online instructions for guidance. Alternatively, you can ask your teachers or your peers.
Try calculating on your calculator. If the answer is exactly , you are in degrees mode. If the answer is about , you are in radians mode.
Your scientific calculator supports the inverse trigonometric functions and (also known as arcsin, arccos and arctan).
Check how to find these functions on your calculator.
For example, on a Casio calculator, press + tan, and ‘’ will appear on the calculator display, ready to use.
Remember to use the correct mode to specify whether your answer will be in degrees or radians.
If you need help using the inverse trigonometric functions, check online for instructions. Alternatively, you can ask your teachers or your peers.
Try calculating on your calculator in degrees mode. The answer should be
Sometimes your calculator will display an error rather than a numerical answer. There are two main types of error:
- Syntax error: This appears if you made an error when typing the expression. You should fix what you typed.
For example, try using your calculator to find “” – which is not a complete expression. This should result in a syntax error.
- Math error: This occurs when a calculation yields no answer, for instance, when dividing by zero. You can interpret this mathematically as an equation having no solution.
For example, try using your calculator to find This should result in a math error, because the maximum value for sine is 1, so 8 is not a valid input.
Choosing the right calculator can make a big difference in your A-levels. Most science students aspiring to medical school also take A-level maths. For these students, a graphic calculator may be a worthwhile investment across multiple subjects, despite its higher cost.
The fx-CG50 from Casio, and its newer model, the fx-CG100, are the most advanced graphic calculators approved for UK A-level exams.
- Best for students seeking to deepen their understanding of concepts, work through complex problems, and explore topics such as vectors and 3D graphing.
The FX-991CW is Casio’s top-tier non-graphic scientific calculator, approved for all major A-level exams.
- Ideal for students who don’t need graphing functions, prefer a familiar layout from GCSE, or want a more affordable yet capable calculator for A-level science and maths.
Disclaimer: Medify Ltd is not affiliated with CASIO or its subsidiaries. References to CASIO products are for educational purposes under fair use.