1.1 Data representation in a computer  (Page 2/6)

For example, 5246 can be expressed like in the following expressions

The place-value of any place to the right of the decimal point is a fraction computed from a product in which the reciprocal of the base—or a fraction with one in the numerator and the base in the denominator—is repeated as a factor exactly as many times as the place is to the right of the decimal point.

For example

Binary system

The binary number system is base 2 and therefore requires only two digits, 0 and 1. The binary system is useful for computer programmers, because it can be used to represent the digital on/off method in which computer chips and memory work.

A binary number can be represented by any sequence of bits (binary digits), which in turn may be represented by any mechanism capable of being in two mutually exclusive states.

Counting in binary is similar to counting in any other number system. Beginning with a single digit, counting proceeds through each symbol, in increasing order. Decimal counting uses the symbols 0 through 9, while binary only uses the symbols 0 and 1.

When the symbols for the first digit are exhausted, the next-higher digit (to the left) is incremented, and counting starts over at 0A single bit can represent one of two values, 0 or 1.Binary numbers are convertible to decimal numbers.

Here's an example of a binary number, ${11101.11}_{(2)}$ , and its representation in the decimal notation

Hexadecimal system

The hexadecimal system is base 16. Therefore, it requires 16 digits. The digits 0 through 9 are used, along with the letters A through F, which represent the decimal values 10 through 15. Here is an example of a hexadecimal number and its decimal equivalent:

The hexadecimal system (often called the hex system) is useful in computer work because it is based on powers of 2. Each digit in the hex system is equivalent to a four-digit binary number. Table below shows some hex/decimal/binary equivalents.

Octal system

Binary is also easily converted to the octal numeral system, since octal uses a radix of 8, which is a power of two (namely, 23, so it takes exactly three binary digits to represent an octal digit). The correspondence between octal and binary numerals is the same as for the first eight digits of hexadecimal in the table above. Binary 000 is equivalent to the octal digit 0, binary 111 is equivalent to octal 7, and so forth.

Converting from octal to binary proceeds in the same fashion as it does for hexadecimal:

And from octal to decimal:

Converting from decimal to base–b

To convert a decimal fraction to another base, say base b, you split it into an integer and a fractional part. Then divide the integer by b repeatedly to get each digit as a remainder. Namely, with value of integer part = $d_{n-1}{d}_{n-2}\mathrm{...}{d}_2{d}_1{d}_{0(10)}$ , first divide value by b the remainder is the least significant digit $a_0$ . Divide the result by b, the remainder is $a_1$ .Continue this process until the result is zero, giving the most significant digit, $a_{n-1}$ . Let's convert ${43868}_{(10)}$ to hexadecimal: