Number system - Amresh Tiwari

By-Amresh Tiwari, Sunbeam Suncity (School & Hostel)

1.
NUMBER
SYSTEM

Contents:-
❑ Definition
❑ Types of Number System
❑ Conversion among Bases
❑ Decimal To Binary Conversion
❑ Binary To Decimal Conversion
❑ Computer Arithmetic or
Binary Arithmetic

Before moving forward let me
share something…

Have you ever
wonder students
how man in pre
historic time’s use
numbers for
counting.

Let me tell there
were animals they
killed, days and
night. But that not
important, what
important is that in
those time man used
his finger to
counting or made
marking on the
bones.

But he could only
go up to 10 on his
hands or up to a
certain limit
using the bones.
For larger numbers
there were no
enough fingers to
use.

So man devised
figures or symbols
representing
numbers and create
a group of symbols
when there was lot to
count.

So here we go to learn
about-
“Number System”

Number system
is a well defined
set of symbols or
digits used to
represent
quantity.
Definition:-

Which number System do we use in
our day to day life?
Ans. Decimal Number System

Types of Number System
On the basis of the symbols/digits
used in a number system, the
Number System divided into 4 Types.
They are…

Number
System
Decimal
Base is 10
(0 - 9)
Binary
Base is 2
(0 - 1)
Octal
Base is 8
(0 - 7)
Hexadecimal
Base is 16
(0 - 9 and A – F)
Note- The BASE or RADIX of a number system is the number of symbols/digits used in it.

Aryabhat was Indian’s
greatest mathematician and astronomer.
He introduced the concept of Zero(0).

Which number System has ‘8’ as its
base?
Ans. Octal Number System

Decimal number system is a well defined set
of 10 symbols or digits used to represent
quantity.
Decimal number system consist of 10 digits,
i.e. 0 to 9 with the base 10.
Digits are -

Binary number system is a well defined set of
2 symbols or digits used to represent
quantity.
Binary number system consist of 2 digits,
i.e. 0 and 1 with the base 2.
Digits are -

Octal number system is a well defined set of 8
symbols or digits used to represent quantity.
Octal number system consist of 8 digits,
i.e. 0 to 7 with the base 8.
Digits are -

Hexadecimal number system is a well defined
set of 16 symbols or digits used to represent
quantity.
Hexadecimal number system consist of 16 digits,
i.e. 0 to 9 and A to F with the base 16.
Digits are -

Conversion among Bases
The possibilities are:-

Binary
Octal
Decimal
Hexadecimal

There are two method:-
Repeated “Division by 2”
Repeated “Find and Subtract”
Conversion Of Decimal into Binary Number System

 Write down the decimal number and divide it by 2 (Base of
Binary Number).
 Write the remainder on the right hand side.
 Divide the result (quotient) of the division by 2 and again
write down the remainder.
 Repeat the dividing process and writing down remainders
until the quotient is 0(zero).
 Write the series of remainder from bottom(Most
Significant Digit) to up (Least Significant Digit). This is the
binary equivalent of the decimal number.
Method 1:- Conversion Of Decimal into Binary
Number System

 Multiply each binary number with its positional value,
which is in terms of power of 2, starting from the
extreme right digit.
 Increase the power one by one, starting from 0(zero),
keeping the base fixed as 2.
 Calculate the sum of all the products to get the
decimal number.
 Write the answer along with its base subscript.
Conversion Of Binary into Decimal Number System

Activity Section
 Convert the following Decimal numbers into Binary
numbers-
A) 68
B) 51
1 1 0 0 1 1
1 0 0 0 1 0 0

For Ex. (25)10 = (?)2
252
6
3
1
12
0
1
0
1
1
2
2
2
2
0
LSD
MSD
Therefore, (25)10 = (11001)2

For Ex. (11001)2 = (?)10
1 1 0 0 1
1 x 20 =
0 x 21 =
0 x 22 =
1 x 23 =
1 x 24 =
1 x 1 =
1 x 8 =
1 x 16 =
0 x 4 =
0 x 2 =
1
16
8
0
0
25
Therefore, (11001)2 = (25)10

 Multiply each octal number with its positional value,
which is in terms of power of 8, starting from the
extreme right digit.
decimal number.
Conversion Of Octal into Decimal Number System

For Ex. (345)8 = (?)10
3 4 5
5 x 80 =
4 x 81 =
3 x 82 =
5 x 1 =
3 x 64 =
4 x 8 =
5
192
32
229
Therefore, (345)8 = (229)10

 Multiply each hexadecimal number with its positional
value, which is in terms of power of 16, starting from
the extreme right digit.
decimal number.
Conversion Of Hexadecimal into Decimal Number
System

For Ex. (345)16 = (?)10
3 4 5
5 x 160 =
4 x 161 =
3 x 162 =
5 x 1 =
3 x 256 =
4 x 16 =
5
768
64
837
Therefore, (345)16 = (837)10

Decimal
To
Binary
Divide by 2
Octal
Divide by 8
Hexadecimal
Divide by 16

Qus : How would you find whether
a number is represented in
Decimal /Binary /Octal or
Hexadecimal number system?
Ans :- With the help of Base or Radix

As a computer understands only the binary code,
so the data entered into the computer by the user is
converted into binary code for processing.
This processing may involve various kinds of
arithmetic operations, such as addition, subtraction,
multiplication, division on the binary numbers.

Binary addition is same as we perform addition with decimal numbers.
Note- While adding 1 + 1, the output will be 10, where 0 is written under the same
column and carry over 1 is shifted to the next place.

Ex. -
11 0
1 1 0 + 1 1 = ?
11+
10
1
1 0
Carry
So, 1 1 0 + 1 1 = 1 0 0 1

Note- The number is borrowed when 1 is subtracted from 0 (10-1=1).

Ex. -
11 0
(1 1 0)2 – (1 1)2 = ?
11-
11
1
0
Borrow
So, (1 1 0)2 – (1 1)2 = (1 1)2
1

Binary multiplication is same as that of the decimal numbers.

Ex. - (1 1 0)2 – (1 1)2 = ?
1 1
1
0
1x
11
11 0
0
01001
So, (1 1 0)2 – (1 1)2 = 1 0 0 1 0

Binary division is same as that of the decimal numbers.
1 1 01 0
Ex. Compute 1 1 0 ÷ 1 0
Divisor
Remainder
Quotient
Dividend
1
1 0
0
1 0
1
1 0
0 0

2
Decimal
8
10
Binary
Hexadecimal
0

F
F
F
F

Similar to ‘Binary to Decimal conversion. The only difference
is the change of base.
The number is borrowed when 1 is subtracted from 0
i.e. 1 0 – 1 = 1

Number system - Amresh Tiwari

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