14.5: Negative Numbers

Last updated
Save as PDF

Page ID: 43747

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

To know the value of a positive number is simple: we just add all the powers of 2 given by the binary representation as explained at the beginning of this chapter. Getting the value of a negative number is quite simple: we do the same except that we count the sign bit as negative and all the other ones as positive. The sign bit is the most significant bit i.e., the bit that represents the largest value (see Figure \(\PageIndex{1}\)). For example, on 8 bit representation it will be the one associated with the weight \( 2^7 \).

Negative numbers in 8 bits. — Figure \(\PageIndex{1}\): Negative numbers on 8 bits.

Let us illustrate that: −2 is represented on 8 bit encoding as: 1111 1110.

To get the value out of the bit representation, we simple add: \( -2^7 + 2^6 + 2^5 + 2^4 + 2^3 + 2^2 + 2^1 + 0 * 2^0 \), i.e., \( -128 + 64 + 32 + 16 + 8 + 4 + 2 \) and we get \( -2 \).

\( -69 \) is represented on 8 bit encoding as: \( 1011 1011 \). To get the value out of the bit representation is simple. We add \( -2^7 + 0 * 2^6 + 2^5 + 2^4 + 2^3 + 0 * 2^2 + 2^1 + 2^0 \), i.e., \( -128 + 32 + 16 + 8 + 2 + 1 \) and we get \( -69 \).

Following the same principle, check that the value of \( -1 \) is the one described in Figure \(PageIndex{1}\).

Let us count a bit: on an 8 bit representation we can then encode 0 to 255 positive integers or -128 to \( 64 + 32 + 16 + 8 + 4 + 2 + 1 \) 127. In fact we can encode from \( -1 * 2^7 \) to \( 2^7 - 1 \). More generally on \( N \) bit we can encode \( -1 * 2^{N-1} - 1 \) integer values.