how IEEE-754 floating point numbers work

Question

Let's say I have this:

float i = 1.5

in binary, this float is represented as:

0 01111111 10000000000000000000000

I broke up the binary to represent the 'signed', 'exponent' and 'fraction' chunks.

What I don't understand is how this represents 1.5.

The exponent is 0 once you subtract the bias (127 - 127), and the fraction part with the implicit leading one is 1.1.

How does 1.1 scaled by nothing = 1.5???

Answer 1

Think first in terms of decimal (base 10): 643.72 is:

• (6 * 10²) +
• (4 * 10¹) +
• (3 * 10⁰) +
• (7 * 10^-1) +
• (2 * 10^-2)

or 600 + 40 + 3 + 7/10 + 2/100.

That's because n⁰ is always 1, n^-1 is the same as 1/n (for a specific case) and n^-m is identical to 1/n^m (for more general case).

Similarly, the binary number 1.1 is:

• (1 * 2⁰) +
• (1 * 2^-1)

with 2⁰ being one and 2^-1 being one-half.

In decimal, the numbers to the left of the decimal point have multipliers 1, 10, 100 and so on heading left from the decimal point, and 1/10, 1/100, 1/1000 heading right (i.e., 10², 10¹, 10⁰, decimal point, 10^-1, 10^-2, ...).

In base-2, the numbers to the left of the binary point have multipliers 1, 2, 4, 8, 16 and so on heading left. The numbers to the right have multipliers 1/2, 1/4, 1/8 and so on heading right.

So, for example, the binary number:

101.00101
| |   | |
| |   | +- 1/32
| |   +---  1/8
| +-------    1
+---------    4

is equivalent to:

4 + 1 + 1/8 + 1/32

or:

    5
5  --
   32

Answer 2

1.1 in binary is 1 + .5 = 1.5

Answer 3

The mantissa is essentially shifted by the exponent.

3 in binary is 0011
3>>1 in binary, equal to 3/2, is 0001.1

Answer 4

You want to read this - IEEE 754-1985

The actual standard is here