Mathematics-1

Lecture-3

Significant Figures

Accuracy

• In the fields of engineering, industry and statistics, the accuracy of a measurement system is the degree of closeness of measurements of a quantity to its actual (true) value.

Precision

• The precision of a measurement system, also called reproducibility or repeatability, is the degree to which repeated measurements under unchanged conditions show the same results.

Low Accuracy High Accuracy High Accuracy

High Precision Low Precision High Precision

Illustration

• The significant figures (also called significant digits and abbreviated sig figs, sign.figs or sig digs) of a number are those digits that carry meaning contributing to its precision.

• Zeroes are sometime used to locate the decimal point, when the zeroes are used in that way we say that they are not significant.

What is a significant figure?

• There are 2 kinds of numbers:

–Exact: the amount of money in your account. Known with certainty.

What is a significant figure?

–Approximate: weight, height—anything MEASURED. No measurement is perfect.

When to use Significant figures

• When a measurement is recorded only those digits that are dependable are written down.

When to use Significant figures

–If you measured the width of a paper with your ruler you might record 21.7cm.

To a mathematician 21.70, or 21.700 is the same.

But, to a scientist 21.7cm and 21.70cm is NOT the same

• 21.700cm to a scientist means the measurement is accurate to within one thousandth of a cm.

Rule-1

• All non-zero digits are considered significant. Example: the number 1 has one significant figure. In 20 and 300, the first figure is significant while the others may or may not be (see below). 123.45 has five significant figures: 1, 2,

3, 4 and 5.. For example, 1 Kg sugar

Rule-2

• Zeros appearing anywhere between two non-zero digits are significant. Example: 101.12 has five significant figures: 1, 0, 1, 1 and 2.

Rule-3

• Leading zeros are not significant. For example, 0.00052 has two significant figures: 5 and 2.

Rule-4

• Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has six significant figures: 1, 2, 2, 3, 0 and 0. The number 0.000122300 still has only six significant figures (the zeros before the 1 are not significant). In addition, 120.00 has five significant figures.

Cont…

• This convention clarifies the precision of such numbers; for example, if a result accurate to four decimal places is given as 12.23 then it might be understood that only two decimal places of accuracy are available. Stating the result as 12.2300 makes clear that it is accurate to four decimal places.

Rule-5

• Alternatively, the above can be summarized by three rules.– All non-zero digits are significant.– In a number without a decimal point, only zeros

BETWEEN non-zero digits are significant (unless a bar indicates the last significant digit--see below).

– In a number with a decimal point, all zeros to the right of the left-most non-zero digit are significant.

Rule-6

• The significance of trailing zeros in a number not containing a decimal point can be ambiguous. For example, it may not always be clear if a number like 1300 is accurate to the nearest unit (and just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundred due to rounding or uncertainty. Various conventions exist to address this issue:

Conventions (Rule-6)

• A bar may be placed over the last significant digit; any trailing zeros following this are insignificant.

• The last significant figure of a number may be underlined; for example, "20000" has two significant figures.

• A decimal point may be placed after the number; for example "100." indicates specifically that three significant figures are meant.

Rule-7

• A number with all zero digits (e.g. 0.000) has no significant digits

How many sig figs?

• 7• 40• 0.5• 0.00003• 7 x 105

• 7,000,000

• 1• 1• 1• 1• 1• 1

How many sig figs here?

• 1.2• 2100• 56.76• 4.00• 0.0792• 7,083,000,000

• 2• 2• 4• 3• 3• 4

How many sig figs here?

• 3401• 2100• 2100.0• 5.00• 0.00412• 8,000,050,000

• 4• 2• 5• 3• 3• 6

Accuracy and Precision AGAIN

• The accuracy of a decimal number is given by the number of significant figures in the number.

• The precision of a decimal number is given by the position of the right most significant digit.

The End

Have Fun Measuring and Happy Calculating!