Numbers

  2. Basics
  3. Mathematics
  4. Numbers
    5. 5'

In physics, data collected from experiments and answers to problems are rarely nice easy whole numbers.

In IB physics, you can no longer simply write a number on the page - there is lots to consider. Think about significant figures, orders of magnitude and standard notation.

Later, we'll get into error and uncertainties too.

Key Concepts

Significant figures

When writing down a value from an experiment, we should generally write the number to the best possible precision for the instrument, for example:

  • 12.2 cm on a ruler
  • -57.0o on a thermometer

Both of these numbers are to 3 significant figures.

In a calculation, we should write our final answer to the smallest number of significant figures provided to us. This is usually 2 or 3 significant figures and will often involve rounding.

To state the number of signficant figures for a given number, follow the appropriate step:

  • For a value of magnitude greater than 1, count the numbers visible to you after any initial zeros: 9.4651 is to 5 sf, 00011 is to 2 sf.
  • For a value of magnitude less than 1, we continue to count the numbers visible to you after any initial zeros: 0.8 is to 1 sf, 0.00011 is to 2 sf.

 

Orders of magnitude

The order of magnitude of a value is its approximate number of digits and is the nearest power of 10.

  • 2 is of order of magnitude 100 (1)
  • 9 is of order of magnitude 10
  • 11 is of order of magnitude 10
  • 86 is of order of magntude 102 (100)
  • The radius of the earth is of order of magnitude 107 m
  • The charge on an electron is of order of magnitude 10-19 C

Note that, unlike rounding on a linear scale (5 or above), the next order of magnitude starts from \(\approx\)4 because of the logarithmic scale.

 

 

Essentials

Mean

When collecting data in an experiment, you might conduct repeated readings. Anomalous values can be discarded and an average can be calculated, which improves the accuracy of the result. In physics, we calculate the mean average:

\(\bar{x}=\frac{x_1+x_2+x_3...+x_n}{n}\)

\(n\) represents the number of readings

 

Standard notation

Because physicists can rarely quote values to more than 2 or 3 signficant figures, very large or very small numbers are expressed in standard notation.

  • \(123 000 000=1.23 \times 10^8\)(count numbers visible after 1st digit)
  • \(0.000321 = 3.21 \times 10^{-4}\)(count zeros after the decimal place and one before - note the minus sign)

 

MY PROGRESS

How much of Numbers have you understood?