How the Mantissa Works

Since leading zeros of a binary number can be dropped, it is safe to not actually include the first 1. There will always be one, so there is no need to show it.

The exponent is adjusted so that the most significant 1 is in the one's place of the mantissa. Then that 1-bit is dropped.

For example: say that you wish to represent 3.17578125 as a float. In binary, this is 00011.00101101.

Drop the leading zeros:
11.00101101
Now there is a leading one.
Move the binary point so that the leading one is in the one's place (the exponent shows the move):
1.100101101 × 2¹
Drop the leading one:
100101101 × 2¹
Add zero bits on the right so that there are 23 bits in total:
10010110100000000000000 × 2¹
Result: the mantissa is: 10010110100000000000000

Answer:

How the Mantissa Works

QUESTION 13: