Appendix A.3: Binary Fractions

Appendix A.3: Binary Fractions#

Floating Point Numbers#

With decimal numbers a decimal point is used to separate the whole and fractional parts of a number. Recalling from Architecture of the Atmel ATmega 328 Microcontroller that we represent numbers using a weighted positional notation in which digit’s value is relative to its position. (Fig. 161) illustrates the idea

We use exponentiation of negative powers of the base to represent the fractional part of a number:

\[14.12_{10} = \left(1\times 10^1\right) + \left(4\times 10^0\right) + \left(1\times 10^{-1}\right) + \left(2\times 10^{-2}\right)\]

The idea extends to binary, octal and hexadecimal numbers:

\[\begin{align*} 0000.101_2 &= \left(1\times 2^{-1}\right) + \left(0\times 2^{-2}\right) + \left(1\times 2^{-3}\right) \\ &= \frac{1}{2} + 0\times \frac{1}{4} + \frac{1}{8}\\ &= 0.5 + 0 + 0.125 = 0.625_{10} \end{align*}\]

The representation of octal and hexadecimal numbers is left as an exercise.

Decimal Floating-Point Number Conversion#

To convert a decimal fraction (base 10) to a new base (n) the fractional part is repeatedly multiplied by n.

The whole number part of the product gives the value at the current power.
The decimal part of the product is multiplied by n and repeated until the fractional part of the product is zero
Read the result from top to bottom

Limitations#

Convert \(0.675_{10}\) to binary

0.675 × 2 = 1.35 1

0.35 × 2 = 0.7 0

0.7 × 2 = 1.4 1

0.4 × 2 = 0.8 0

0.8 × 2 = 1.6 1

0.6 × 2 = 1.2 1

0.2 × 2 = 0.4 0