# Tutorial 2 - Using Standard Form and Significant Figures

 Contents

Standard Form

Standard form consists of a number between 1 and 10 multiplied by a power of 10.  For big numbers and very small numbers standard form is very useful.

 Comment on what happens if you try to put the following numbers into your calculator as they are. Can you do any calculations on them? (a) 3200 (b) 5 600 000 (c) 2 800 000 000 000 (d) 0.000000000000341

You should have found that very small numbers entered into a calculator are read as 0, unless they are entered as standard form.

The following number is shown in standard form:

3.28 × 105

= 3.28 × 100 000 = 328 000

Consider this number:

We find that there are 18 digits after the first digit, so we can write the number in standard form as:

4.505 × 1018

For fractions we count how far back the first digit is from the decimal point:

0.00000342

In this case it is six places from the decimal point, so it is:

3.42 × 10-6

A negative power of ten (negative index) means that the number is a fraction, i.e. between 0 and 1.

 Convert the following numbers into standard form: 86; 381; 45300; 1 500 000 000;  0.03;  0.00045;  0.0000000782

There is no hard and fast rule as to when to use standard form in an answer.  Basically if your calculator presents an answer in standard form, then use it.  I generally use standard form for:

• numbers greater than 100 000

• numbers less than 0.001

When doing a conversion from one unit to another, for example from millimetres to metres, I consider it perfectly acceptable to write:

15 mm = 15 × 10-3 m

 Avoid using expressions like 1 billion.  It is pure journalese.     1 billion can mean 1 thousand million (1 × 109) or 1 million million (1 × 1012).

Using a Calculator

A scientific calculator is an essential tool in Physics, just like a chisel is to a cabinet-maker.  A calculator geared just to money is fine for an accounts clerk, but quite useless to a physicist.  All physics exams assume you have a calculator, and you should always bring a calculator to every lesson.  They are not expensive, so there is no excuse for not having one.  It is rather depressing when a student fails to bring a calculator and says, "Forgorrit."

The calculator should be able to handle:

• standard form;

• trigonometrical functions;

• angles in degrees and radians;

• natural logarithms and logarithms to the base 10.

Most scientific calculators have this and much more.

There are no hard and fast rules as to what calculator you should buy:

• Get one that you are happy with.  I used to use an ancient thing that is nearly thirty years old, but it worked.  Then I lost it.

• Make sure it is accurate; I have known some calculators to get an answer plain wrong!  This usually happens when the batteries are failing.

• Avoid machines that need a hefty instruction manual.

• For the exam, there are certain types of calculator that are NOT allowed, for example those with QWERTY keypads.  Make sure that your calculator is an allowable type.

Some Points

I am assuming that you know the basic functions of your calculator, but I need to draw your attention to a couple of points:

Do NOT write 2.317.  This means "2.31 to the power 7" (= 351).  This will get a mark deduction because it's an arithmetical error.

Different calculators use slightly different ways of keying in inputs.  You need to be careful with the use of brackets.

Consider this calculation:

The correct way to do this is:

This is wrong:

There is no excuse for such calculator errors.

Significant Figures

Suppose we have a measurement of 5 m.  We could say that it’s 5.0 m, 5.00 m, or 5.000 m.  All of these suggest a difference in precision:

• 5 m suggests a precision of the nearest metre;

• 5.00 m suggests a precision of the nearest cm;

• 5.000 m suggests a precision of the nearest mm.

To find the number of significant figures, you have to count the total number of digits, starting at the first non-zero digit.  0.000034 is to two significant figures.

Rounding is done in the usual way that you will have done in Maths. So 4.73 is rounded to 4.7 to 2 significant figures, and 5 to 1 significant figure.

 Do not confuse significant figures with places of decimals.      2.3 is two significant figures but one decimal place.    0.0023 is to two significant figures but four decimal places.                          0.0023 = 0.00 to two decimal places.

Consider this calculation:

Vrms = 13.6

Ö2

There is no reason at all in A-level Physics to write any answer to any more than 3 significant figures.  Three significant figures is claiming accuracy to about one part in 1000.  Blindly writing your calculator answer is claiming that you can be accurate to one part in 100 million, which is absurd.  Before calculators were common, slide rules were used extensively.  They would give answers to two significant figures, three at a push.

The examination mark schemes give answers that are no more than 2 significant figures.  So our answer becomes:

Vrms = 9.62 V (3 s.f.)

Vrms = 9.6 V (2 s.f.)

Do any rounding up or down at the end of a calculation.  If you do any rounding up or down in the middle, you could end up with rounding errors.

Many questions tell you to write your answer to an appropriate number of significant figures.  The rule is:

The answer should be to the same number of significant figures as the quantity with the lowest number of significant figures.

For a question that quotes these data:

h = 6.63 × 10-34 Js

e = 1.6 × 10 -19 C

m = 9.11 × 10 -31 kg

V = 1564 V

The answer will be to no more than 2 significant figures, as the quantity e is to 2 significant figures.

Some other tips on use of calculators:

• On most calculators the number is keyed in before the function (sin, cos, log)

• Take one step at a time and write intermediate results.

• It is easy to make a mistake such as pressing the × key rather than the ÷ key.  It is a good idea to do the calculation again as a check.

As you get more experienced, you will get a feel for what is a reasonable answer. 1000 N is a reasonable force that a car would use to accelerate; 2 × 10-10 N is most certainly not.

 Use your calculator to do the following calculations.  Write your answers to no more than three significant figures. (a) 3.4 × 10-3 × 6.0 × 1023          235   (b) 27.32 – 24.82            Ö38   (c) 1.45093   (d) sin 56.4   (e) cos-1 0.4231   (f) tan-1 2.143   (g) sin-1 1.00052   (h) Reciprocal of 2.34 × 105   (i) log10 200   (j) 45 sin 10

Note that if you have a formula that contains an integer (a whole number), such as:

You ignore the integer as far as significant figures are concerned.

Mathematicians often write answers in surd form, e.g. 5Ö8.  You need to express an answer like this as a proper number, 14.1 (to 3 significant figures) in this case.

Calculators may express a number as a fraction, e.g.:

245

37

Again, this needs to be written as a decimal number:

6.62 (to 3 significant figures)

 A runner runs a 100.0 m race in a time of 13 s. Calculate his average speed, giving your answer to an appropriate number of significant figures.