Number System  Aptitude basics, practice questions, answers and explanations 
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Number Systems

When we consider a number in a decimal system we can divide it into units, tens, hundreds, one tenth, one hundredth etc. For example the numeral 572.65 can be written as (5*10²) + (7*10¹) + (2*10⁰⁾ + (6*10^-1) + (5*10^-2).

We say that “10” is the base of the number system.

Base

The number which decides the place value of a symbol or a digit in a number.  Alternatively, it is the number of distinct symbols that are used in that system. The base should be a positive integer other than 1. If N is any integer, r is the base of the system and a₀, a₁, a₂… a_n be the digits required to present N, then

N= a_nrⁿ+a_n-1r^n-1+ ……. +a₁r+a₀, where 0≤a_i≤r-1

Eg: (i) (143)₅ = 1*5² + 4*5¹ + 3*5⁰ = 48

(ii) (1101)₂ = 1*2³ + 1*2² + 0*2¹ + 1*2⁰ = 13.

Note:  The subscript indicates the base. In the above examples 5 and 2 are bases. We can also represent fractions in other bases. For example (0.572)₈ = 5*1/8 + 7*1/8² + 2*1/8³.

The following table lists some number systems along with their base and symbols.

A=10, B=11, C=12, D=13, E=14, F=15, some books denote ten as “E” and eleven as “e”.

The conversion of a number from one base to the other and the arithmetic operations involving bases other than 10 are discussed in worked out examples.

We need to remember the elementary rules while adding binary numbers.

1.Convert (216.42)₈ into base 10.

Sol. (216.45)8 = 2*8² +1*8¹ + 6*8⁰ + 4*8^-1 + 2*8^-2

= 128+8+6+ ½ + 1/32 = (142.53125)10

2.Convert (1101.11)2 into base 10.

Sol. (1101.11)2 = 1*23 + 1*22 + 0*21 + 1*20 + 1*2-1 + 1*2-2

= 8+4+1+ ½ + ¼ = (13.75)10

3.Convert (456)₁₀ into base 8.

4.Convert (27)10 into base 2.

Thus (27)10  =(11011)2

5.Find (1100101)2 + (110)2

6.Find (101110)2 + (111011)2