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BASE NUMBER SYSTEM
The number system. is used for reading, writing, counting and calculation. It is based on some characters called digits. Each number is made up of these digits.

Base of a Number System
The number of digits a system uses is called its base. The present system of counting and computation is base ten number system.This is called decimal number system.

(i) Binary Number System (Number system with base 2)
In base two system, the smallest numeral is 0 and the greatest numeral is 1. In decimal system, the greatest numeral is 9, in which when '1' is added it becomes 10 (read as ten). In the same sense when '1' is added to '1' in base two system, we get two written as '10' and read as 'one-zero' and not ten.

Thus, in binary system: one is written as 1₂

Now, 1₂+1₂=One plus one is Two which is written as 10₂.

Again, 10₂+1₂=Two plus one is Three which is written as 11₂

11₂+1₂=Three plus one is Four which is written as 100₂

Note: In binary system, any number can be expressed as the sum of multiples of powers of 2.

For example:

13=8+4+1=1x2³+1x2²+0x2¹+1x2⁰ =1101₂

27=16+8+2+1=1x2⁴+1x2³+0x2²+1x2¹+1x2⁰=11011₂

(ii) Number system with base 5
The base 5 number system is based on the five fundamental digits 0, 1, 2, 3, and 4 to represent the numeric values.The smallest numeral value is 0 and greatest numeral value is 4.

The smallest numeral is 0 and the greatest numeral is 4.

In decimal system 9+1= 10, but in base five system 4₅+1₅ =Five which is written as 10₅.

Similarly, 10₅+1₅=11₅ (Six)

Now,

11₅+1₅=12₅ (Seven)

Again 12₅+1₅=13₅ (Eight)

And 13₅+1₅=14₅ (Nine), 14₅+1₅=20₅ (Ten) and 20₅+1₅=21₅ (Eleven)

Note: Any number in base 5 can be expressed as sum of multiples of powers of 5.

For example:

(a) 87=75+10+2=3x25+2×5+2×1

Or        87=3×5²+2×5¹+2×5⁰

87=75+10+2=3x5²+2x5¹+2x 5⁰=322₅

(b) 138=125+10+3=1x125+0x25+2x5+3x1

Or        138=1x5³+0x5²+2x5¹+3x5⁰=1023₅

(iii) Octal Number System (Number system with base 8)
The number system with base 8 consists of the following eight fundamental digits 0, 1, 2, 3, 4, 5, 6 and 7 to represent the numeric value.The smallest numeral value is 0 and greatest numeral value is 7 to represent the numeric values.

Note: Any number in base 8 can be expressed as sum of multiples of powers of 8.

For example:

135=128+7=2x64+0x8+7×1

Or     135 = 2x8²+0x8¹+7x8⁰=207₈

(iv) Decimal Number System (Number system with base 10)
The number system with base 10 is also called decimal system. Decimal number system is most familiar to us. We use this decimal number system in our daily life and business for counting and calculations. In decimal system, we count in tens using the digits : 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

Note: In decimal number system, any integer can be expressed as sum of powers of 10.

For example:

5683= 5x1000+6x100+8x10 +3

Or   5683= 5x10³+6x10²+8x10¹+3x10⁰

== Conversion ==

(i). Conversion form decimal to binary
We use successive division method. In this method the given number is repeatedly divided by 2, till the remainder is 1. Then arranging the remainder numbers from bottom to get required number in the binary system.

Example: Convert 276 into binary system.

Solution:

276/2=138 (Remainder=0)

138/2= 69 (Remainder=0)

69/2= 34 (Remainder=1)

34/2= 17 (Remainder=0)

17/2= 8 (Remainder=1)

8/2= 4 (Remainder=0)

4/2= 2 (Remainder=0)

2/2=1 (Remainder=0)

1/2=0 (Remainder=1)

276=(100010100)₂

(ii). Conversion from decimal to base 5 system
We use same division method, and given number is repeatedly divided by 5, till the dividend is 4 or less then 4. Then arranging the remainder numbers from bottom to get the required number in base 5 system

Example: Convert 6065 into base five system Solution:

6065/5=1213 (Remainder=0)

1213/5=242 (Remainder=3)

242/5= 48(Remainder=2)

48/5= 9 (Remainder=3)

9/5= 1 (Remainder=4)

1/5=0(Remainder=1)

6065=(143230)₅

(iii). Conversion from decimal to octal
we use same division method, the given number is repeatedly divided by 8, till the dividend is 7 or less then 7. Then arranging the remainder numbers from bottom to get the required number in octal system

Example: Convert 2064 into octal system.

Solution:

2064/8=258 (Remainder=0)

258/8=32 (Remainder=2)

32/8= 4(Remainder=0)

4/8= 0 (Remainder=4)

2064=(4020)₈

(i). Conversion from binary to decimal
Example: Convert (110101)₂ into decimal.

Solution:

110101₂=1×2⁵+1×2⁴+0×2³+1×2²+0×2¹+1×2⁰

=1×32+1×16+1×4+1×1

=32+16+4+1=53

(110101)₂=53

(ii). Conversion from base 5 system to decimal
Example: Convert (234)₅ into decimal.

Solution:

234₅=2×5²+3×5¹+4×5⁰

=2×25+3×5+4×1

=50+15+4=69

(234)₅=69

(iii). Conversion from octal to decimal
Example: Convert (1456)₈ into decimal.

Solution:

1456₈=1×8³+4×8²+5×8¹+6×8⁰

=1×512+4×64+5×8+6×1

=512+256+40+6=814

(1456)₈=814