Accuracy of numbers

(1) Approximate numbers.There are two types of numbers: exact and approximate. Exact numbers are 2, 4, 9, , , 6.45, ...... etc. but there are numbers such that (= 1.333 ......), 2 (= 1.414213 ...) and π (= 3.141592......) which cannot be expressed by a finite number of digits. These may be approximated by numbers 1.3333, 1.4141, and 3.1416, respectively.

Such numbers, which represent the given numbers to a certain degree of accuracy, are called approximate numbers.

(2) Significant digits.The digits used to express a number are called significant digits.

The digits 1, 2, 3, 4, 5, 6, 7, 8, 9 are significant digits. ‘0’ is also a significant digit except when it is used to fix the decimal point or to fill the places of unknown or discarded digits.

For example, each of the numbers 7845, 3.589, and 0.4758 contains 4 significant figures while the numbers

0.00386, 0.000587, 0.0000296

contain only three significant figures (since zeros only help to fix the position of the decimal point).

Similarly, in the number 0.0003090, the first four ‘0’ s’ are not significant digits since they serve only to fix the position of the decimal point and indicate the place values of the other digits. The other two

‘0’ s’ are significant.

To be more clear, the number 3.0686 contains five significant digits.

A.The significant figure in a number in positional notationconsists of

(i) All non-zero digits

(ii) Zero digits which

(a) lie between significant digits;

(b) lie to the right of decimal point and at the same time to the right of a non-zero digit;

B.The significant figure in a number written in scientific notation(e.g., M × 10k) consists of all the digits explicitly in M.

Significant digits are counted from left to right starting with the nonzero digit on the left.

A list is provided to help students understand how to calculate significant digits in a given number:

(3) Rounding-off.There are numbers with many digits, e.g., = 3.142857143. In practice, it is desirable to limit such numbers to a manageable number of digits, such as 3.14 or 3.143. This process of dropping unwanted digits is called rounding-off.

Numbers are rounded-off according to the following rule:

To round-off a number to n significant digits, discard all digits to the right of the nth digit and if this discarded number is

(i) less than 5 in (n + 1)th place, leave the nth digit unaltered. e.g., 7.893 to 7.89.

(ii) greater than 5 in (n + 1)th place, increase the nth digit by unity, e.g., 6.3456 to 6.346.

(iii) exactly 5 in (n + 1)th place, increase the nth digit by unity if it is odd, otherwise leave it unchanged. e.g., 12.675 ~ 12.68

12.685 ~ 12.68

The number thus rounded-off is said to be correct to n significant figures.

A list is provided for explanatory proposes:

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