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 adusted so that the most significant 1 is in the one's place of the mantissa. For example: say that you wish to represent 00011.00101101 as a float.

• 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