Maths - using units, decimals and standard form

Significant figures

Sometimes we do not always need to give detailed answers to problems - we just want a rough idea. When we are faced with a long number, we could round it off to the nearest thousand, or nearest million. And when we get a long decimal answer on a calculator, we could round it off to a certain number of decimal places. Another method of giving an approximated answer is to round off using significant figures.

The word "significant" means "having meaning".

With the number 368249, the 3 is the most significant digit, because it tells us that the number is 3 hundred thousand and something. It follows that the 6 is the next most significant, and so on.

With the number 0.0000058763, the 5 is the most significant digit, because it tells us that the number is 5 millionths and something. The 8 is the next most significant, and so on.

Be careful with numbers such as 30245, as the 3 is the first significant figure and 0 the second, because of its value as a place holder.

We round off a number using a certain number of significant figures. The most common are 1, 2 or 3 significant figures.

Remember the rules for rounding up are the same as before:

  • if the next number is 5 or more, we round up
  • if the next number is 4 or less, we do not round up

When using significant figures in calculations, we need to take into account the measurements we have made.

If measurements have been made to one, two or three significant figures, we cannot have more significant figures in answers to any calculations we make.

Using units

Most animal and plant cells are 0.01 to 0.10 mm in size. The smallest thing seen with the naked eye is about 0.05 mm.

For all cells we need a microscope to see them in any detail.

The best unit to measure most cells is the micrometre, which has the symbol μm.

For some sub-cellular structures, for instance ribosomes, or organisms such as viruses, it’s best to use a smaller unit – the nanometre.

One metre can be broken down into the following measurements:

Millimetre, mmMicrometre, μmNanometre, nm
\frac{1}{1000}\; metre \frac{1}{1000}\; millimetre \frac{1}{1000}\; micrometre
Division of a metre as a fraction \frac{1}{1000}\; metre \frac{1}{1\: 000\: 000}\; metre \frac{1}{1\: 000\: 000\: 000}\; metre
Division of a metre in standard form1 × 10-3 m1 × 10-6 m1 × 10-9 m

A unit we use in everyday lives is the centimetre, \frac{1}{100} m, or 1 × 10-2 m.

Decimals

Decimals allow us to clearly identify that a number is not whole. They show us fractions of numbers in a very clear way.

So 0.5 is halfway between 0 and 1. 0.25 is one quarter of the way and 0.75 is three quarters.

  • 0.1 is a tenth (1 ÷ 10)
  • 0.01 is a hundreth (1 ÷ 100)
  • 0.001 is a thousandth (1 ÷ 1000)

Standard form - Higher

When writing and working with very large or very small numbers, we use standard form.

Standard form shows the size of numbers as powers of 10.

curriculum-key-fact
Standard form numbers are written as:
  • A × 10n
  • A is a number greater than 1 but less than 10
  • n is the index or power, always in powers of 10

Using standard form for large numbers

  • A population of 120 000 000 microorganisms could be written as 1.2 × 108.
  • This number can be written as 120 000 000.0.
  • If the decimal place is moved eight spaces to the left we get 1.2.
  • So we put × 108 after 1.2 to show this.
  • It makes a very large number easier to write down.

Using standard form for small numbers

  • A red blood cell's diameter of 7 μm or 0.000007 m could be written as 7 × 10-6 m.
  • This number can be written as 0.000007.
  • If we move the decimal place six spaces to the right we get 7.0.
  • So we put × 10-6 after 7 to show this.
  • Because the original number is less than 1 metre, we put a minus sign before the 6.
  • It makes a very small number easier to write down.

Calculating the magnification of a cell

Image of dividing cells and their measurements

In a book, a micrograph of the cell measured 100 mm.

The real size of the cell shown is 0.05 mm.

To calculate the magnification:

\textup{magnification} = \frac{100\; \textup{mm}}{0.05\; \textup{mm}} = 2000

It's important to work in the same units when calculating magnification. Sizes of most cells are given in micrometres, with the symbol μm.

To calculate magnification using the same formula in micrometres, convert the measurement of the cell above from millimetres (mm) into micrometres (μm):

Cell measurement = 100 mm

1 mm = 1000 μm

100 mm = 100 × 1000 μm

100 mm = 100 000 μm

The real size of the cell above in micrometres is 50 μm.

The magnification of the image:

\textup{magnification} = \frac{100\; 000 \;  \textup{μm}}{50\;   \textup{μm}} = 2000

From this, we know that the image has been magnified 2000 times.