Mathematics for A-level Science

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Mathematics for A-level Science Units and prefixes Mathematics for A-level Science Copyright © AQA and its licensors. All rights reserved.

Why are units and prefixes important? One of reasons we use the international system of units is because it makes the conversion of units (especially those with different prefixes) mathematically simple. We use prefixes as shorthand for standard form when using commonly occurring very large or very small numbers. This makes it easier to discuss and talk about sets of these numbers. For example, the length 0.0000000023 m may be written as 2.3 × 10–9 m 2.3 is the digit number and is kept. 10–9 is known as the exponential number and can be replaced with the prefix ‘n’ pronounced as ‘nano’. Hence 0.0000000023 m = 2.3 × 10–9 m = 2.3 nm Copyright © AQA and its licensors. All rights reserved.

Why are units important? When measuring a variable in science we must consider the best unit (and prefix) to use. This often differs depending on the subject. Variable Prefix Unit Example Length centi (c) metre (m) 15 cm Volume milli (m) Litre (L) 330 mL Force kilo (k) Newton (N) 0.8 kN Area centi metre squared 624 cm 2 Version 3.0 Copyright © AQA and its licensors. All rights reserved.

Prefixes Here are the common unit prefixes you are likely to encounter: Number Exponential number Prefix billion 1 000 000 000 109 G ‘giga’ million 1 000 000 106 M ‘mega’ thousand 1000 103 k ‘kilo’ tenth 0.1 10–1 d ‘deci’ hundredth 0.01 10–2 c ‘centi’ thousandth 0.001 10–3 m ‘milli’ millionth 0.000 000 1 10–6 μ ‘micro’ billionth 0.000 000 000 1 10–9 n ‘nano’ Copyright © AQA and its licensors. All rights reserved.

Examples Example 1 The length of a DNA nucleotide is 0.6 nm. Convert this number into standard form. b) If a strand of DNA is 1.6 m long, how many nucleotides is it made up of? Copyright © AQA and its licensors. All rights reserved.

Examples Example 1 The length of a DNA nucleotide is 0.6 nm. Convert this number into standard form. 0.6 nm = 0.6 × 10–9 m b) If a strand of DNA is 1.6 m long, how many nucleotides is it made up of? Copyright © AQA and its licensors. All rights reserved.

Examples Example 1 The length of a DNA nucleotide is 0.6 nm. Convert this number into standard form. 0.6 nm = 0.6 × 10–9 m b) If a strand of DNA is 1.6 m long, how many nucleotides is it made up of? 1.8 m / 0.6 nm = 1.8 m / (0.6 × 10–9 m) = 3 × 109 = 3 billion Copyright © AQA and its licensors. All rights reserved.

Manipulating units A number and a unit (like 3 m) is a magnitude (3) multiplied by a unit (metre). The rules of algebra apply not only to the numbers you are manipulating, but also to the units attached to them. For example: 3 m × 3 m = 3 × 3 × m × m = 9 × m × m = 9 × m2 = 9 m2 Units can be multiplied and divided just like regular number. For example: 6 m 3 2 m 2 =3 m 3 m 2 =3 m × m ×m m ×m =3 m × m ×m m ×m =3 m Copyright © AQA and its licensors. All rights reserved.

Manipulating units At A-level, rather than write m/s to mean metres per second, we will write ms–1. This makes it easier to combine units via the following rules: 𝑢𝑛𝑖𝑡 𝑎 × 𝑢𝑛𝑖𝑡 𝑏 = 𝑢𝑛𝑖𝑡 𝑎+𝑏 𝑢𝑛𝑖𝑡 𝑎 𝑢𝑛𝑖𝑡 𝑏 = 𝑢𝑛𝑖𝑡 𝑎−𝑏 Nb This means that: 1 kg −1 =kg or more generally 1 𝑎 −𝑛 = 𝑎 𝑛 For more information about these properties see the indices lesson. Copyright © AQA and its licensors. All rights reserved.

Examples Example 2 a) Calculate the following: 36 cm 3 12 cm 2 b) Calculate the following: 36 kg cm −3 64 cm −2 Copyright © AQA and its licensors. All rights reserved.

Examples Example 2 a) Calculate the following: 36 cm 3 12 cm 2 36 cm 3 12 cm 2 =3 cm 3−2 =3 cm 1 =3 cm  b) Calculate the following: 36 kg cm −3 64 cm −2   Copyright © AQA and its licensors. All rights reserved.

Examples Example 2 a) Calculate the following: 36 cm 3 12 cm 2 36 cm 3 12 cm 2 =3 cm 3−2 =3 cm 1 =3 cm  b) Calculate the following: 36 kg cm −3 64 cm −2   36 kg cm −3 64 cm −2 =0.5 kg cm −3 cm −2 =0.5 kg cm −3−(−2) =0.5 kg cm −3+2 =0.5 kg cm −1 Copyright © AQA and its licensors. All rights reserved.

Exam paper: Example Copyright © AQA and its licensors. All rights reserved.

Exam paper: Example 1 𝟗𝟎 𝐜𝐦 𝟑 𝐤𝐠 −𝟏 𝐡𝐨𝐮𝐫 −𝟏 means 90 cm 3 of oxygen required per kg per hour. Therefore, 90 cm 3 kg −1 hour −1 ×0.4 kg=36 cm 3 hour −1 of oxygen required 𝟕 𝐜𝐦 𝟑 𝐝𝐦 −𝟑 means 7 cm 3 of oxygen absorbed per dm 3 of water passing over gills Therefore 36 cm 3 hour −1 7 cm 3 dm −3 =5.1 dm 3 hour −1 When we multiply 90 by 0.4 the kg and ‘per kg’ cancel out leaving a volume of oxygen required for the specific mass of the fish. 7 cm^3 per dm^3 looks complex mathematically, but it just details the volume of absorbable oxygen in every dm^3 of water. When we divide 36 by 7 the cm^3 and the cm^3 cancel each other out, leaving units of dm^3 per hour. (Remember 1/(x^-1) = x) As the question asks for the oxygen required in one hour our answer becomes 1 x 5.1 dm^3 hour^-1 = 5.1 dm^3. Copyright © AQA and its licensors. All rights reserved.