Introduction to binary math for dummies:

It is a method of writing numbers by using two digits ,0 and 1.The 'binary system' of numeration was used early in the seventeenth century by Thomas Harriot.Each successive bit position in binary number reading from right to left has place value of 1,2,4,8,16 and so on.Binary systems are used in digital electronics. Many electronics instruments are designed using the concept of the binary systems. In this topic we will learn to write the binary numbers and also binary maths like binary addition, binary subtraction, binary multiplication and binary division.

 

Example:- The numbers 0 to 10 are represented in binary system as:

0  --> 0                       (0x1=0)

1-->01                       (1x1+0x2=1)

2-->10                       (0x1 +1x2=2)

3-->11                       (1x1 +1x2=3)

4-->100                     (0x1 +0x2 +1x4=4)

5-->101                    (1x1 +0x2 +1x4=5 )

6-->110                    (0x1+1x2+1x4= 6  )

7-->111                    (1x1 +1x2 +1x4 =7 )

8-->1000                  (0x1 +0x2 +0x4 +1x8=8 )

9-->1001                   (1x1 +0x2 +0x4+1x8=9  )

10-->1010               (0x1 +1x2 +0x4 +1x8=10  )

Octal system:

It is a number system based on base 8,that represents any number as the sum of successive powers of 8.

Example:

 Number in octal system                       Equivalent number in decimal system
20                      (0x80 + 2x81) 16
25                     (5x80 + 2x81 ) 21
71                     (1x8+ 7x81                        57
173                   ( 3x80 + 7x81 + 1x82 ) 123

Hexadecimal System:

Hexadecimal system is a system of numeration with base 16 and requires ciphers from 0 to 15,i.e, 0 to 9 and A to F. where

A =10             B=11                             C=12                               D=13                                    E=14                               F=15

Example: The hexadecimal numbers 5BC3 can be expressed as:

5 x163  + 11x162  + 12x161  + 3x160  and is equivalent to decimal number 23491.

 

Binary Addition and substraction in math for dummies:

 

Binary Addition:

The binary numbers are added as in simple addition.whenever a carry results in addition it is prefixed at the start of the result at its M.S.B. or most significant bit position.

Example:      

01  (1)

10   (2)

             101   (5)

              011    (3)

Answer:    11   (3)

Carry bit 1000    (8)

1011     (11)

0100     ( 4)

                1100

                0111

1111     (15) carry bit   10011    (19)

Binary Subtraction:

The subtraction of binary numbers is done in the 2-complement notation and then adding the numbers in order to reduce complexity. if a carry results in the M.S.B. position, it is ignored.

Process: The first number is left as it is and the second number i.e the subtrahend is represented in the 2-complement notation. Finally, the process of addition  is carried out.

example:    1001

           -        0111

0111 is represented as     1000

                                              +      1

Now                            1001               which is the 2- complemented notation of 0111.

                               +1001

                                     (1)    0010(2)

Hence , 0010 which is in 2 in its binary form is the desired answer.

 

Binary Multiplication and Division in math for dummies

 

Binary Multiplication:

The multiplication process of binary numbers is carried out as in the simple multiplication.

Example:               1000      (8)

                   x            0110       (6)

                   -------------------------------

                               0000

                             1000

                           1000

                          0000

                    -----------------------

                         0110000  (48)

                  ----------------------------

Binary Division:

The division process of binary numbers is carried out in simple division while the binary numbers are subtracted in their 2-complemented notation.

Process:

Firstly, the most significant bits of divisor and dividend are compared to initiate the division process. If the M.S.B. of dividend is less than that of divisor, thent the next most significant bit is put and division process is started and continues till the end as in the normal division.

         110 )  110110 (

                   -110

            ------------------------

                              110

                            -110

           -----------------------

                              x

          ---------------------------  Hence , quotent =1001 (9), remainder =0. 

 

 

Practice problems on binary math

 

     a)   11100                b)                1011                c)                101                        d)       111 )  111111(   

        +  10101                              -    0011                                 x111

     --------------------                 ------------------------                ------------------

Answers:   a) 110001             b) 1000                             c)100011                      d) 1001