1/ Combinations of SI Base Units
To make working with units easier, some combinations of the base units are given special names, but it is always correct to reduce everything to the base units. Table 2.2 lists some examples of combinations of SI base units that are assigned special names. Do not be concerned if the formulae look unfamiliar at this stage - we will deal with each in detail in the chapters ahead (as well as many others)!

It is very important that you are able to recognise the units correctly. For instance, the newton (N) is another name for the kilogram metre per second squared (kg·m·s$^{−2}$), while the kilogram metre squared per second squared (kg·m²·s$^{−2}$) is called the joule (J).

Important: When writing combinations of base SI units, place a dot (·) between the units to indicate that different base units are used. For example, the symbol for metres per second is correctly written as m·s$^{−1}$, and not as ms$^{−1}$ or m/s.

2/ Scientific Notation
In Science one often needs to work with very large or very small numbers. These can be written more easily in scientific notation, in the general form

d × 10$^{e}$​

where d is a decimal number between 0 and 10 that is rounded off to a few decimal places. e is known as the exponent and is an integer. If e > 0 it represents how many times the decimal place in d should be moved to the right. If e < 0, then it represents how many times the decimal place in d should be moved to the left. For example 3,24 × 10³ represents 3240 (the decimal moved three places to the right) and 3,24 × 10$^{−3 }$represents 0,00324 (the decimal moved three places to the left).

If a number must be converted into scientific notation, we need to work out how many times the number must be multiplied or divided by 10 to make it into a number between 1 and 10 (i.e. the value of e) and what this number between 1 and 10 is (the value of d). We do this by counting the number of decimal places the decimal comma must move. For example, write the speed of light in scientific notation, to two decimal places. The speed of light is 299 792 458 m·s$^{−1}$. First, find where the decimal comma must go for two decimal places (to find d) and then count how many places there are after the decimal comma to determine e.

In this example, the decimal comma must go after the first 2, but since the number after the 9 is 7, d = 3,00. e = 8 because there are 8 digits left after the decimal comma. So the speed of light in scientific notation, to two decimal places is 3,00 × 10⁸m·s$^{−1}$
3/ Significant Figures

In a number, each non-zero digit is a significant figure. Zeroes are only counted if they are between two non-zero digits or are at the end of the decimal part. For example, the number 2000 has 1 significant figure (the 2), but 2000,0 has 5 significant figures. You estimate a number like this by removing significant figures from the number (starting from the right) until you have the desired number of significant figures, rounding as you go. For example 6,827 has 4 significant figures, but if you wish to write it to 3 significant figures it would mean removing the 7 and rounding up, so it would be 6,83.

Exercise: Using Significant Figures​

E - 1. Round the following numbers:
(a) 123,517 ℓ to 2 decimal places
(b) 14,328 km·h$^{−1}$ to one decimal place
(c) 0,00954 m to 3 decimal places

Hướng dẫn

1. a) 123,58ℓ
2. b) 14,3 km·h$^{−1}$
3. c) 0,01 m

E - 2. Write the following quantities in scientific notation:
(a) 10130 Pa to 2 decimal places
(b) 978,15 m·s$^{−2}$ to one decimal place
(c) 0,000001256 A to 3 decimal places

Hướng dẫn

(a) 10130 Pa = 10⁵ Pa
(b) 0,98 m/s²
(c) 0,000001256 A = 1,25.10^-6 A

E - 3. Count how many significant figures each of the quantities below has:
(a) 2,590 km
(b) 12,305 mℓ
(c) 7800 kg

Hướng dẫn

(a) 2,6 km
(b) 12 mℓ
(c) 7800 kg

4/ Prefixes of Base Units
Now that you know how to write numbers in scientific notation, another important aspect of units is the prefixes that are used with the units.

Definition: Prefix
A prefix is a group of letters that are placed in front of a word. The effect of the prefix is to change meaning of the word. For example, the prefix un is often added to a word to mean not, as in unnecessary which means not necessary.

In the case of units, the prefixes have a special use. The kilogram (kg) is a simple example. 1 kg is equal to 1000 g or 1 × 10³g. Grouping the 10³ and the g together we can replace the 10³ with the prefix k (kilo). Therefore the k takes the place of the 10³. The kilogram is unique in that it is the only SI base unit containing a prefix. In Science, all the prefixes used with units are some power of 10. Table 2.4 lists some of these prefixes. You will not use most of these prefixes, but those prefixes listed in bold should be learnt. The case of the prefix symbol is very important. Where a letter features twice in the table, it is written in uppercase for exponents bigger than one and in lowercase for exponents less than one. For example M means mega (10⁶) and m means milli (10$^{−3}$).

Important: There is no space and no dot between the prefix and the symbol for the unit.
Here are some examples of the use of prefixes:

• 40000 m can be written as 40 km (kilometre)
• 0,001 g is the same as 1 × 10$^{−3}$ g and can be written as 1 mg (milligram)
• 2,5 × 10⁶N can be written as 2,5 MN (meganewton)
• 250000 A can be written as 250 kA (kiloampere) or 0,250 MA (megaampere)
• 0,000000075 s can be written as 75 ns (nanoseconds)
• 3 ×10$^{−7}$mol can be rewritten as 0,3 ×10$^{−6}$mol, which is the same as 0,3 µmol (micromol)

Exercise: Using Scientific Notation​

E - 1. Write the following in scientific notation using Table 2.4 as a reference.
(a) 0,511 MV
(b) 10 cℓ
(c) 0,5 µm
(d) 250 nm
(e) 0,00035 hg

Hướng dẫn

(a) 0,511 MV = 0,511 × 10⁶V
(b) 10 cℓ = 10^-3 ℓ
(c) 0,5 µm = 5.10^-7 m
(d) 250 nm = 250 × 10^-9m
(e) 0,00035 hg = 0,035 g

E - 2. Write the following using the prefixes in Table 2.4.
(a) 1,602 × 10$^{−19}$C
(b) 1,992 ×10⁶J
(c) 5,98 ×10⁴N
(d) 25×10$^{−4}$A
(e) 0,0075×10⁶m

Hướng dẫn

(a) 1,602 × 10$^{−19}$C = 0,1602 aC
(b) 1,992 ×10⁶J = 1,992 MC
(c) 5,98 ×10⁴N = 59,8kN
(d) 25×10$^{−4}$A = 2,5mA
(e) 0,0075×10⁶m = 7,5km

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Grade 10 - Physics - Units

High School Students Studying the Sciences Physics