We depend on the concept of "POSITIONAL NOTATION" wherein the position of digits in a representation implies an association with a "POSITIONAL VALUE" - commonly called the Hindu-Arabic number system.

Contrast with the "Additive Notation" used in the Roman number system where each character represented a value, and the sum of the value representations determined the whole number.

**Decimal Representation of integers**

- In base 10 (decimal) notation, a positive integer
*x***value**is uniquely**represented**by*k*+1 decimal digits*d*_{k}d_{k-1}...d_{1}*d*_{0}

*d*is in the set {1,2,3,4,5,6,7,8,9}_{k}

*d*(belongs to) {0,1,2,3,4,5,6,7,8,9}, i=0 ,....,_{i}*k*-1and the value associated with the representation can be computed by the formula:

*x*=*d*10_{k}^{k}+ d_{k-1}10^{k- 1}+....+*d*_{1}10^{1}+*d*_{0}10^{0} -
**Examples:**

143_{10}= 1·100 + 4·10 + 3·1

5261_{10}= 5·10^{3}+ 2·10^{2}+ 6·10^{1}+ 1·10^{0}

**Binary Representation of integers**

- In base 2 (binary) notation, a positive integer
*x*is uniquely__represented__by*k*+1 binary digits*d*_{k}d_{k-1}...d_{1}*d*_{0}

*d*= 1_{k}

*d*(belongs to) { 0 , 1 }, i=0 ,....,_{i}*k*-1and the value associated with the representation can be computed by the formula:

*x*=*d*2_{k}^{k}+ d_{k-1}2^{k- 1}+....+*d*_{1}2^{1}+*d*_{0}2^{0} -
**Examples:**

13_{10}= 1·8 + 1·4 + 0·2 + 1·1

= 1101_{2}

61_{10}= 1·32 + 1·16 + 1·8 + 1·4+0·2 + 1·1

= 111101_{2} - Alternatively notice that
*x*=*d*_{0}+ 2(*d*_{1}+2(*d*_{2}+.....+2(*d*_{k-1}+2*d*_{k})...))We will use both of these value expressions in developing the algorithms for number conversion.

**NOTES:**- A normalized representation is does not include any "leading zeros" and thus the leftmost digit is always non-zero. In the case of a binary number system the leading digit is always 1, in decimal can be 1, ..., 9.
However is a representation that uses a fixed number of digits (such as "8-bit") then it is necessary to add leading zeros to the left of the leftmost non-zero digit.
- If you look at the formulae for defining positional notations above you will find that it is quite feasible to have negative base values. There is quite a wealth of literature in Computer Science regarding the advantages of using a number system with a base of -2!

**QUESTION:**How many zeros are there to the left of the leftmost non-zero digit in ANY positional notation?

CS1104 Main Page

Last Updated 2001/09/17

© L.Heath, 2000, modified by J.A.N. Lee.

- A normalized representation is does not include any "leading zeros" and thus the leftmost digit is always non-zero. In the case of a binary number system the leading digit is always 1, in decimal can be 1, ..., 9.
However is a representation that uses a fixed number of digits (such as "8-bit") then it is necessary to add leading zeros to the left of the leftmost non-zero digit.