Question

Int is 4 bytes with a range of +- 2^31
Float is 4 bytes with a range of +- 1.2E(+- 38)

Float encompasses so many more points on the Real Line and yet is equal to the size of int. Is the Sign-Exponent-Fraction representation of float so awesome (or the 2’s complement of Int so pathetic) that this size disparity arises? Am I missing something?

I just find it highly surprising that something which represents (virtually) the entire real line is of the same size as that of which represents the Integers.

Answer 1

In C, an int and float each take up 4 bytes, or 32 bits, in memory. The difference between int and float is not the number of bits they take up in memory, but rather in the way the ALU (Arithmetic Logic Unit) treats each number. An int is treated as the integer represented by its bits using two’s complement notation. A float, on the other hand, is encoded (typically in IEEE 754 format) to represent a number in exponential form (e.g. 2.9979245×10⁸ is a number in exponential form with base 2.9979245 and exponent 8). The number of significant digits in the decimal representation of a floating point number is always about the same: 6-9 digits for 32-bit float and 15-17 digits for a 64-bit float (or double). For example, the value 4.2949673×10⁹ is what I get as the decimal representation of the closest 32-bit float to the number 2³² , and it has 8 significant digits.

It can be a common misconception that a floating point number somehow represents more information than a fixed-point number, which is not true. While a 32-bit float can represent numbers of greater magnitude than a 32-bit int, it cannot represent all of them with as much precision. In total, this answer derives that a 32-bit float can represent around 4,278,190,081 unique numbers. This other answer contains a nice plot showing how these 32-bit floats are distributed on the real line.

While an int may not be able to represent numbers with as large of a magnitude as a float, it represents each number in its valid range (-2³² to 2³²-1) with the same precision. The largest number it can represent is 2,147,483,647 and has 10 significant digits. And in total, it can represent 2³²=4,294,967,296 unique numbers, which is more than for the 32-bit float!