Floating Point Binary Numbers Computer Science — Student Guide
Floating Point Binary Numbers
How computers store decimal numbers using just 0s and 1s
Computers can only store 1s and 0s (binary digits). That's fine for whole numbers like 42 or 255 —
but what about 6.75 or −0.001? This document explains the trick computers use: floating point
representation.
1. Quick Revision — Binary & Powers of Two
You already know that binary whole numbers work like this:
Binary Calculation Decimal value
1101 1×8 + 1×4 + 0×2 + 1×1 13
1010 1×8 + 0×4 + 1×2 + 0×1 10
11111111 128+64+32+16+8+4+2+1 255
The same idea extends to the right of the binary point (like a decimal point, but for binary). Each
position to the right is half the previous one:
Position Value
1s place (2⁰) 1
½s place (2 ¹)⁻0.5
¼s place (2 ²)⁻0.25
⅛s place (2 ³)⁻0.125
1/16s place (2 ⁴)⁻0.0625
So 101.11 in binary means: 4 + 0 + 1 + 0.5 + 0.25 = 5.75 in decimal. Easy! The challenge is
storing this efficiently using a fixed number of bits.
2. The Three Parts of a Floating Point Number
The most common floating point standard is IEEE 754 single precision, which uses exactly 32
bits split into three fields:
Sign Exponent Mantissa (Fraction)
1 bit 8 bits 23 bits
Sign (1 bit): 0 = positive, 1 = negative. Simple!
Exponent (8 bits): Controls how big or small the number is — like the power in scientific notation.
A bias of 127 is added before storing, so the stored value is always positive.
