DCLD : Number System - Converting from other number bases to decimal

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Important Questions: DCLD - Page 4

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Converting from other number bases to decimal

Other number systems use different bases. The binary number system uses base 2, so the place values of the digits of a binary number correspond to powers of 2. For example, the value of the binary number 10011 is determined by computing the place value of each of the digits of the number:

1	0	0	1	1	the binary number
2^4	2^3	2^2	2^1	2^0	place values

So the binary number 10011 represents the value

	(1 * 2^4)	+	(0 * 2^3)	+	(0 * 2^2)	+	(1 * 2^1)	+	(1 * 2^0)
=	16	+	0	+	0	+	2	+	1
=	19

The same principle applies to any number base. For example, the number 2132 base 5 corresponds to

2	1	3	2	number in base 5
5^3	5^2	5^1	5^0	place values

So the value of the number is

	(2 * 5^3)	+	(1 * 5^2)	+	(3 * 5^1)	+	(2 * 5^0)
=	(2 * 125)	+	(1 * 25)	+	(3 * 5)	+	(2 * 1)
=	250	+	25	+	15	+	2
=	292

Converting from decimal to other number bases

In order to convert a decimal number into its representation in a different number
base, we have to be able to express the number in terms of powers of the other base. For example, if we wish to convert the decimal number 100 to base 4, we must figure out how to express 100 as the sum of powers of 4.

100	=	(1 * 64)	+	(2 * 16)	+	(1 * 4)	+	(0 * 1)
	=	(1 * 4^3)	+	(2 * 4^2)	+	(1 * 4^1)	+	(0 * 4^0)
Then we use the coefficients of the powers of 4 to form the number as represented in base 4:
100	=	1 2 1 0	base 4

One way to do this is to repeatedly divide the decimal number by the base in which it is to be converted, until the quotient becomes zero. As the number is divided, the remainders - in reverse order - form the digits of the number in the other base.

Example: Convert the decimal number 82 to base 6:

82/6	=	13	remainder 4
13/6	=	2	remainder 1
2/6	=	0	remainder 2

The answer is formed by taking the remainders in reverse order: 2 1 4 base 6

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