BINARY NUMBER SYSTEM - PLACE VALUE CHART, CONVERSION TO AND FROM BASE-10 SYSTEM

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Please study Number Systems before Binary Number System,
if you have not already done so.

There we have seen base-10 number system
or decimal number system which is the most popular system
used by humans throughout the world.

But, computers work internally with only two symbols
because of the straightforward implementation
in digital electronic circuitry using logic gates.

Thus, the base-2 number system is the basis for digital computers.
It is used to perform integer arithmetic in almost all digital computers.

The two base symbols or digits used in binary system are
0 called zero and 1 called one. We are already familiar with
these symbols or digits in decimal number system.

Let us learn how to write numbers using the binary number system.

This system is analogous to the decimal number system
in following the place value rule.

There, value of the place becomes ten times,
as we move one place to the left.and here it
becomes two times.

Place value rules in Binary number System

The value of the right extreme place is one (1) or unity.

Value of the place increases as it moves to the left.

Value of the place becomes two times, as we move one place to the left.

So

the value of the place second from right is two times one and is equal to two.

The value of the place third from right is two times two and is equal to four.

The value of the place, fourth from right is two times four and is equal to eight.

The value of the place, fifth from right is two times eight and is equalto sixteen.

Thus, the next place values are thirty two, sixty four,
one hundred twenty eight and so on.

Conversion of base-two numerals into base-ten numerals

The following examples will make the process clear.

Example 1 Binary number System

The following table shows the place value chart for 1001
in the binary number system.

Eights' place Fours' place Twos' place Units' place
1001

In the above example, the value of 1001
= 1 eights + 0 fours + 0 twos + 1 ones
= 8 + 0 + 0 + 1 = 9
It is written as one-zero-zero-one.

Let us see how the single digit numbers in decimal number system
are represented in binary number system.













Representation of single digit numbers
of decimal number system in binary number system

Decimal SystemBinary System
00
11
210
311
4100
5101
6110
7111
81000
91001

Note :
(i) subscript (2) indicates that the number is in binary number system.
(ii) where no subscript (base) is shown, it should be taken as 10.

Example 2 Binary number System

Write 10010(2) in the decimal number system.

Solution :

168421
1 0010

The value of 10010(2)
= 1(16) + 0(8) + 0(4) + 1(2) + 0(1)
= 16 + 0 + 0 + 2 + 0 = 18. Ans.

Example 3 of Binary number System

Write 1110011(2) in the decimal system.

Solution :

64 32168421
1 110011

The value of 1110011(2)
= 1(64) + 1(32) + 1(16) + 0(8) + 0(4) + 1(2) + 1(1)
= 64 + 32 + 16 + 0 + 0 + 2 + 1 = 115. Ans.





Conversion of base-ten numerals into base-two numerals

We use division method. We successively divide by 2 and take
the remainder 0 or 1 in successive places starting from units' place.
We continue the process till the quotient is 0.

The following examples will make the process clear.

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Example 4 of Binary number System

Write 36 in the binary system.

Solution :


2 | 36
------
2 | 18  -  0     Units' place
------
2 |  9  -  0     Twos' place
------
2 |  4  -  1     Fours' place
------
2 |  2  -  0     Eights' place
------
2 |  1  -  0     Sixteens' place
------
  |  0  -  1     Thirty two's place

Thus, 36 = 100100(2)











Example 5 of Number Systems : Binary System

Write 101 in the binary system.

Solution :


2 | 101
-------
2 |  50  -  1     Units' place
-------
2 |  25  -  0     Twos' place
-------
2 |  12  -  1     Fours' place
-------
2 |   6  -  0     Eights' place
-------
2 |   3  -  0     Sixteens' place
-------
2 |   1  -  1     Thirty two's place
-------
  |   0  -  1     Sixty four's place

Thus, 101 = 1100101(2)

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Example 6 of Binary number System

Write 1227 in the binary system.

Solution :


2 | 1227
--------
2 |  613  -  1    units' place
--------
2 |  306  -  1    Twos' place
--------
2 |  153  -  0    Fours' place
--------
2 |   76  -  1    Eights' place
--------
2 |   38  -  0    Sixteens' place
--------
2 |   19  -  0    Thirty twos' place    
--------
2 |    9  -  1    Sixty Fours' place
--------
2 |    4  -  1    One hundred twenty eights' place
--------
2 |    2  -  0    Two hundred fifty six' place
--------
2 |    1  -  0    Five Hundred twelve's place
--------
  |    0  -  1    One thousand twenty four's place

Thus, 1227 = 10011001011(2)

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Exercise of Binary number System

(1) Write the following numbers in decimal system.

(i) 1001(2) (ii) 11100(2)(iii) 1000101(2)

(iv) 110101(2)(v) 10001110(2) (vi) 1110001111(2)

(2) Write the following numbers in binary system.

(i) 47 (ii) 89 (iii) 721 (iv) 123 (v) 2049 (vi) 1123

For answers see at the bottom of the page.

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Answers to Exercise of Binary number System

(1) (i) 9 (ii) 28 (iii) 69 (iv) 53 (v) 142 (vi) 911

(2) (i) 101111 (ii) 1011001 (iii) 1011010001 (iv) 1111011

(v) 100000000001 (vi) 10001100011