Number system

Category : 9th Class

*      Number system

 

The number of digits in the system determines the base of any number system, such as decimal and binary. Basically binary is a base -2 number system as it uses two basic digits. Thus it means that the whole Binary number system depends on the two basic digits. Whereas Decimal is a base-10 system since it uses ten digits and these are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.  

 

* Decimal Number System

A system with base - 10 is a decimal number system. Thus it means that there are ten basic digits on which the decimal number system depends. The digits are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. By using these ten digits all the numbers in decimal number system are written. Thus the place value of a digit in a number increases the power from right to left.  

 

* The following are the place value of each digit of number 5471 : 

  • The place value of 1 is:  \[1*{{10}^{0}}=1\]
  • The place value of 6 is : \[7*{{10}^{1}}=70\]
  • The place value of 3 is : \[4*{{10}^{2}}=400\]
  • Theplacevalueof5  is : \[5*{{10}^{3}}=5000\]

 

 

*  Binary Number System

A number system with a base-2 is known as binary number system. The whole binary number system depends on two digits these are 0 and 1, respectively. By using these two digits the number in binary number system are written. Thus the place value of a digit in a number increases in the power of 2 from right to left.

 

* The following example shows how to convert binary number 1010101 into decimal number:

Power of

2

6

5

4

3

2

1

0

Binary number

1

0

1

0

1

0

1

 

 

  • The place value of 1 is : \[1*{{2}^{0}}=1\] 
  • The place value of 0 is : \[0*{{2}^{1}}=0\]
  • The place value of 1 is : \[1*{{2}^{2}}=4\] 
  • The place value of 0 is :\[0*{{2}^{3}}=0\]
  • The place value of 1 is : \[1*{{2}^{4}}=16\]
  • The place value of 0 is : \[0*{{2}^{5}}=0\]
  • The place value of 1 is : \[1*{{2}^{6}}=64\]
  • The decimal number = \[1+4+16+64=85\]  

 

 

* Decimal to Binary Conversion

While converting decimal to binary there are two methods. These are:

  • Comparing with descending powers of two and subtraction
  • Short division by two with remainder

Comparing with descending powers of two and subtraction:  

 

* The following example shows all the steps 

\[{{156}_{10}}\]

 

  • The greatest power of 2 that fits into 156 = 128
  • So write 1 for the leftmost binary digit
  • Binary digit =1
  • Subtract 128 from 156, that is 156-128 =28
  • Now check the next lower power of 2. It is 64 but 64 does not fit into 28. So write 0 for the next binary digit to right.
  • Binary digit = 10
  • Now check the next lower power of 2. It is 32 but 32 does not fit into 28. So write 0 for the next binary digit to right.
  • Binary digit = 100
  • Now check the next lower power of 2. It is 16 and fits into 28. Subtract 16 from 28.
  • 28-16=12
  • So write 1 for the next binary digit to right.
  • Binary digit = 1001
  • Now check the next lower power of 2. It is 8 and fits into 12. Subtract 12 from 8.
  • 12 - 8 = 4
  • So write 1 for the next binary digit to right.
  • Binary digit = 10011
  • Now check the next lower power of 2. It is 4 and fits into 4. Subtract 4 from 4.
  • 4 - 4 = 0
  • So write 1 for the next binary digit to right.
  • Binary digits 100111
  • Now check the next lower power of 2. It is 2 and does not fit into 0.
  • So write 0 for the next binary digit to right.
  • Binary digit = 1001110
  • Now check the next lower power of 2. It is 1 and does not fit into 0.
  • So write 0 for the next binary digit to right.
  • Binary digit = 10011100 Short division by two with remainder: In this method divide each new quotient by 2 and write the reminders to the right of each dividend.  

 

* The following example shows the method:

2) 312   0

2) 156   0

2) 78     0

2) 39     1

2) 19     1

2) 9       1

2) 4       0

2) 2       0

2) 1       1

0

Binary digit = 100111000  

 

 

*  Hexadecimal Number System

The hexadecimal number system is based on 16. Therefore, it means, there are 16 basics digits on which whole hexadecimal number system depends. The digits are 0,1,2,3,4,5,6,7,8,9,10,11,12, 13,14 and 15, where as the numbers 10,11,12,  13, 14, and 15 are also represented as A, B, C, D, E and F. By using these 16 digits all the numbers in Hexadecimal number system are written. Thus the place value in hexadecimal system is increased in the power of 16 from right to left.  

 

* The following example shows the place value of each digit in the number 121 FA: 

  • Place value of A is : \[10*{{16}^{0}}=10\]
  • Place value of F is : \[15*{{16}^{1}}=240\]
  • Place value of 6 is : \[1*{{16}^{2}}=256\]
  • Place value of 2 is : \[2*{{16}^{3}}=8192\]
  • Place value of 1 is : \[1*{{16}^{4}}=65536\]

 

* One's Complement

One's component is a system that is used to represent negative numbers. To take 1s complement of binary digit, replace all 1s with 0s and all 0s with 1s.      

 

 

example.jpg  

1 s complement of 110001 is 001110.    

 

 

Question.jpg    

 

 Steve wants to convert 10111 to decimal number. Which one of the following is the correct conversion?

(A) 48                                                                   

(B) 23

(C) 29                                                                   

(D) 3000

(E) None of these

 

Answer: (B)  

Explanation

Correct Option:

(B) The place value of 1 is : \[1*{{2}^{0}}=1\]

The place value of 0 is : \[1*{{2}^{1}}=2\]

The place value of 1 is : \[1*{{2}^{2}}=4\]

The place value of 0 is : \[0*{{2}^{3}}=0\]

The place value of 1 is : \[1*{{2}^{4}}=16\]

The decimal number =\[1+2+4+0+16=23\]

Incorrect Options:

Rest of the options is incorrect.    

 

 

 Find out the one's complement of 1100111?

(A) 0011000                                       

(B) 0011111

(C) 1100110                                        

(D) 0100110

(E) None of these

 

Answer: (A)  

Explanation  

Correct Option:

(A) The one's complement of 1100111 is 0011000.  

Incorrect Options:

Rest of the options is incorrect.    

 

 

 The hexadecimal number system is based on 16. Which one of the following is an example of hexadecimal number?

(A) 232G                                                             

(B) 137H

(C) 120AG                                                           

(D) 121BC

(E) None of these

 

Answer: (D)  

Explanation  

Correct Option:

(D) The digits are 0,1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14 and 15, whereas the numbers 10,  11, 12,13, 14, and 15 are also represented as A, B,C, D, E and F.  

Incorrect Options:

Rest of the options is incorrect.



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