Paper 4: Fundamentals of Business Mathematics & Statistic

2.28 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Algebra

Finding the Characteristic

(i) Let the number be greater than 1.

Any number whose integral part is of one digit only lies between 1 and 10. For example 5.7 lies between

1 and 10.

Log 5.7 lies between 0 and 1.

Now log 1 = 0 and log 10 = 1.

i.e. log 5.7 = 0 + a proper fraction.

Hence the characteristic of such numbers is 0. Again any number consisting of two digits in the integral

part lies between 10 and 100.

i.e., 11, 60.7, 75.1, 92.9 etc.

Now log 11 lies between 1 and 2 (Q log 10 = 1, log 100 = 2)

i.e. log 11 = 1 + a positive proper fraction.

Hence the characteristic of such numbers is 1.

Similarly, any number consisting of three digits in the integral part lies between 100 and 1000. Therefore,

its logarithm lies between 2 and 3.

Hence the characteristic of such numbers is 2. Thus we arrive at following rule :

Rule 1 : The characteristic of the logarithm of a number greater than 1 is positive and is less by one than the

number of the digits in the integral part.

Thus the characteristics of log 234, log 2.34 are respectively 2, 0.

(ii) Let the number of less than 1 (but greater than 0) i.e., a decimal fraction.

100 = 1 ∴ log 1 = 0

10 –1 = 0.1 ∴ log (0. 1) = – 1

10 –2 = 0.01 ∴ log (0. 01) = – 2

10 –3 = 0.001 ∴ log (0. 001) = – 3 and so on.

Any fraction lying between 0. 1 and 1 has no zero between the decimal point and the first significant digit.

For example 0.31, 0.471.

Now 0.31 lies between 0.1 and 1.

i.e., log 0.31 lies between log 0.1 and log 1.

∴ log 0.31 lies between – 1 and 0.

i.e. log 0.31 = – 1 + a positive proper fraction.

Hence the characteristic of such numbers is – 1.

Again any fraction lying between 0.01, 0.1 has one zero between the decimal point and the first significant

digit. For example, 0.031, 0.047, 0.0707 etc.

Now 0.031 lies between 0.01 and 0.1

or log 0.031 lies between – 2 and – 1.( ∴ log 0.01 = – 2 log 0.1 = – 1)

∴ log 0.031 = – 2 + a positive proper fraction.

Hence the characteristic of such numbers is – 2.

Similarly, any fraction lying between 0.001 and 0.01, has two zeros between the decimal point and first

significant digit. For example 0.0031, 0.0047 etc.