## Mantissa

10. Properties of the Mantissa:

1. The mantissa is same for the same order of digits in two different numbers, irrespective of where the decimal point is in the two numbers

Consider the two numbers:

log 256 = 2.408 = 2 + 0.408

log 25.6 = 1.408 = 1 + 0.408

log 2.56 = 0.408 = 0 + 0.408

so, what do we observe from the logarithmic values on the right of the three numbers on the left.

We see, the characteristic differs, but the mantissa i.e., the proper positive fraction in the sums above is same though the numbers 256, 25.6 and 2.56 are different and differ in the position of the decimal point.

The reason?

Since the three numbers have the same digits namely 2, 5 and 6 and in the same order 2, 5, 6therefore the (Positive Proper Fraction)0.408 is same for all the three numbers.

2. The Mantissa is always written positive

The characteristic of the logarithm of a number can be either positive or negative.

But the Mantissa of the logarithm of a number is always positive

Consider the following examples:

log 0.5 = -0.301 = -1 + 0.699 = ¯1+ 0.699
log 0. 08 = -1.096 = -2 + 0.904 = ¯2 + 0.904
log 0.005 = -2.3010 = -3 + 0.699 = ¯3+0.699

In each of the above three examples, we see that the negative logarithmic values (in the center) of the numbers (at left) are converted into a sum of a negative integer (the characteristic) and a positive proper fraction

So, the mantissa is always written as a positive number i.e., a positive proper fraction.
Furthermore, to indicate that the Mantissa is never negative and it is characteristic that can be negative, we write a bar on the characteristic as shown in the three examples above. The bar tells the logarithm of the number is negative and so it is the characteristic that is converted into a negative integer.

11. How to find the Mantissa :

The Mantissa of the logarithm of numbers is found using logarithm tables.
In finding the Positive Proper Fraction of logarithms of numbers, we consider the following two points:

1. the significant digits in a number, i.e., the digits other than 0
2. 0 also when it occurs in between the significant digits.
Zeroes occurring either after or before the significant digits are ignored while finding mantissa.

For example in the number 0.00456, only the number 456 is taken into account, while the zeroes

before the significant digits 4, 5 and 6 are ignored.
But in the number, 45.006, the two zeroes after the decimal point are not ignored, rather taken into account for finding the mantissa.

12. What is Antilogarithm

Consider the example:

log a x = n
we know n is the logarithm of x to base a.
We also say that x is the antilogarithm of n to base a.
We know that
log 10 15 = 1.176
so 1.176 is called the antilog of 15 to base 10.
Antilogarithms are found using antilog tables