1 Obtain a scientific calculator and practise using it:learn to use scientific notation with the calculator, e.g. to enter a number in the form 3.5 x 106;ensure the angle mode is set to degrees, not radians or gradient;practise using functions such as log, sin, cos and sin-1.
2 If you require further assistance with maths, you might like to look at websites designed for this purpose.
Mathematics - a brief review
Equations
An equation is a mathematical statement in which the left and right sides are equal.
It can contain numbers, symbols and functions. Symbols are letters standing for numbers (for example x, y). It is common to use a standard symbol for a particular physical quantity (e.g. f for the ultrasound frequency) but this is purely a convention. Suffixes can be used to allow the same letter to be used for related quantities (e.g. c1 and c2 to represent the propagation speed in the first and second tissue).
Examples of functions are trigonometric functions like sine, cosine and tangent (abbreviated sin, cos and tan respectively). Note that 5y means the same as (5 × y), i.e. 5 multiplied by y.
An equation can be rearranged so that you can evaluate one quantity as a function of all the others. This is achieved by adding or subtracting the same amount from both sides or multiplying or dividing each side by the same amount (all of which leave the equation still true). The left and right hand sides can also be interchanged. For example:
and:
Addition, subtraction, multiplication and division
If several things are added and/or subtracted (or multiplied and/or divided) together, it doesn't matter in which order this is done. For example:
and
An exception is when some of the equation is in brackets – in this case do the part of the calculation inside the brackets first.
Units
Units specify how a particular quantity is measured. Tables 1.1 and 1.2 list the standard physical units for measurements.
Table 1.1 Units for fundamental quantities.
| Quantity | Units | Abbreviation |
|---|---|---|
| mass | kilogram | kg |
| length | metre | m |
| time | second | s |
| electric current | Ampere | A |
| temperature | degree Celsius | °C |
Table 1.2 Units for derived quantities.
| Quantity | Units | Abbreviation |
|---|---|---|
| energy | Joule | J |
| power | Watt | W |
| intensity | Watt/cm2 | W/cm2 |
| pressure | Pascal | Pa |
| velocity | metre/sec | m/s |
| frequency | Hertz | Hz |
It is common to use prefixes to modify these units to make them more convenient. For example, length may be measured in metres, centimetres or millimetres (abbreviated m, cm and mm) and time may be measured in seconds, milliseconds or microseconds (sec, msec, μsec). Table 1.3 defines the meaning of the commonly used prefixes.
Table 1.3 Common prefixes for units.
| Quantity | Abbreviation | Meaning |
|---|---|---|
| giga | G | × 109 |
| mega | M | × 106 |
| kilo | k | × 103 |
| centi | c | × 10-2 |
| milli | m | × 10-3 |
| micro | μ | × 10-6 |
Whenever you are making a calculation, make sure the units are consistent (e.g. if one length is in mm and one in cm, convert one of them so they are both in the same units).
You should also look at the units you are using to determine the units that your answer will be in. Write the units after each number or symbol, and multiply and divide units just as if they were numbers.
For example, you are asked to calculate
where ap and λ are both measured in mm. Then
Dividing top and bottom by mm gives
and so the answer will be in mm.
If you want to change units, once again just write the units as if they were numbers. For example, if λ = 0.3 cm and we want to express it in mm instead, we simply replace [cm] with [10 mm] (since 1 cm = 10 mm) and then take the 10 out of the square brackets and multiply by it:
Scientific notation
When dealing with very large and very small numbers, it is useful to use scientific notation. This consists of a number between 1 and 10 multiplied by the appropriate power of 10. Working with this notation has several advantages. It is