THE BINARY NUMBER SYSTEM [LECTURE I]

THE COMPUTER NUMBER SYSTEM

Today we are starting a new topic after successfully rounding up the previous topic of logarithm, and i will be starting with the Binary numbers before moving into other forms of number systems in our future lectures.

BINARY NUMBERS
The system of numbers in everyday use is the denary or decimal
system of numbers, using the digits 0 to 9. It has ten different
digits $(0, 1, 2, 3, 4, 5, 6, 7, 8$ and $9)$ and is said to have a radix or
base of $10$.
The binary system of numbers has a radix of 2 and uses only
the digits $0$ and $1$.

Conversion of Binary to Denary

The denary number $234.5$ is equivalent to
$2\times{10^2}+3\times{10^1}+4\times{10^0}+5\times{10^{-1}}$

i.e. is the sum of terms comprising: (a digit) multiplied by (the
base raised to some power).

   In the binary system of numbers, the base is $2$, so $1101.1$ is
equivalent to: $1\times 2^3 + 1\times 2^2 + 0 \times 2^1 + 1\times 2^0 + 1\times 2^{−1}$
Thus the denary number equivalent to the binary number
$1101.1$ is
$8+4+0+1+\frac{1}{2}$, that is $13.5$.

i.e. $1101.12=13.5_{10}$, the suffixes $2$ and $10$ denoting binary and
denary systems of numbers respectively.

Example 1: Convert $11011_2$ to a denary number.
$11011_2=1\times 2^4+1\times 2^3+0\times 2^2+1\times 2^1+1\times 2^0$
$=16+8+0+2+1$
$=27_{10}$

Example 2: Convert $0.1011_2$ to a decimal fraction
$0.1011_2=1\times 2^{-1}+0\times 2^{-2}+1\times 2^{-3}+1\times 2^{-4}$
$=1\times\frac{1}{2}\times 0\times\frac{1}{2^2}+\frac{1}{2^3}+1\times\frac{1}{2^4}$
$=\frac{1}{2}+\frac{1}{8}+\frac{1}{16}$
$=0.5+0.125+0.0625$
$=0.6875_{10}$