The sign bit is as simple as it gets. 0 denotes a positive number; 1 denotes a negative number. Flipping the value of this bit flips the sign of the number.
The exponent field needs to represent both positive and negative exponents. To do this, abias is added to the actual exponent in order to get the stored exponent. For IEEE single-precision floats, this value is 127. Thus, an exponent of zero means that 127 is stored in the exponent field. A stored value of 200 indicates an exponent of (200-127), or 73. For reasons discussed later, exponents of -127 (all 0s) and +128 (all 1s) are reserved for special numbers.
For double precision, the exponent field is 11 bits, and has a bias of 1023.
The mantissa, also known as the significand, represents the precision bits of the number. It is composed of an implicit leading bit and the fraction bits.
To find out the value of the implicit leading bit, consider that any number can be expressed in scientific notation in many different ways. For example, the number five can be represented as any of these:
5.00 × 100
0.05 × 102
5000 × 10-3
In order to maximize the quantity of representable numbers, floating-point numbers are typically stored innormalized form. This basically puts the radix point after the first non-zero digit. In normalized form, five is represented as 5.0 × 100.
A nice little optimization is available to us in base two, since the only possible non-zero digit is 1. Thus, we can just assume a leading digit of 1, and don't need to represent it explicitly. As a result, the mantissa has effectively 24 bits of resolution, by way of 23 fraction bits.