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
1
0
0
1
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 System
Binary System
0
0
1
1
2
10
3
11
4
100
5
101
6
110
7
111
8
1000
9
1001
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 :
16
8
4
2
1
1
0
0
1
0
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
32
16
8
4
2
1
1
1
1
0
0
1
1
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.
Get The Best Grades With the Least Amount of Effort
Here is a collection of proven tips, tools and techniques to turn you into a super-achiever - even if you've never thought of yourself as a "gifted" student.
The secrets will help you absorb, digest and remember large chunks of information quickly and easily so you get the best grades with the least amount of effort.
If you apply what you read from the above collection, you can achieve best grades without giving up your fun, such as TV, surfing the net, playing video games or going out with friends!
Progressive Learning of Math : Binary Number System
Recently, I have found a series of math curricula (Both Hard Copy and Digital Copy) developed by a Lady Teacher who taught everyone from Pre-K students to doctoral students and who is a Ph.D. in Mathematics Education.
This series is very different and advantageous over many of the traditional books available. These give students tools that other books do not. Other books just give practice. These teach students “tricks” and new ways to think.
These build a student’s new knowledge of concepts from their existing knowledge. These provide many pages of practice that gradually increases in difficulty and provide constant review.
These also provide teachers and parents with lessons on how to work with the child on the concepts.
The series is low to reasonably priced and include