the Decimal System

The Decimal System

·         Comprises 10 digits: 0-9, which are recognized as the symbols in the system.

·         Each digit represents units ten times the unit of the digits to its right.

·         Take(125)₁₀ for example:
125 = 5(10⁰) + 2(10¹) + 1(10²)

·         Which gives you  5 + 20 + 100 = 125.

Conversion from Decimal to Binary

Again taking the number 125 for example:

(a) Divide the decimal number by 2.

(b) Write the remainder (which is either 0 or 1) at the right most position.

(c) Repeat the process of dividing by 2 until the quotient is 0 and keep writing

the remainder after each step of division.

(d) Write the remainders in reverse order.

*However, this method is tedious when you use it with a large number, say 9542, therefore not recommended.

Alternatively, we can also:

1.       we know for a fact that 125 is less than 128, which is 2⁷.

2.       Therefore the value would hold 7 bits _ _ _ _ _ _ _₂

3.       Work it out this way:

4.       125 – 2⁶ = 61

5.       61 – 2⁵ = 29 ->  1 1 _ _ _ _ _₂

6.       29 – 2⁴ = 13 ->  1 1 1 _ _ _ _₂

7.       13 – 2³ = 5 =>   1 1 1 1 _ _ _₂

8.       4 + 1 = 5 = 1(2²) + + 0(2¹) + 1(2⁰) -> 1 1 1 1 1 0 1₂

Converting Decimal Fractions to Binary

Take 0.75 for example.

1.       Multiplying the decimal fraction by 2, you’d get 0.75 x 2 = 1.50

2.       If a non-zero integer is generated, record the non-zero integer. Otherwise, record 0.

3.       Remove the non zero integer, and repeat the procedures until the fraction value becomes zero.

4.       In this case, we would get

0.75 x 2 = 1.50	1
0.50 x 2 = 1.00	1

5.       Therefore (0.75)₁₀ would be (0.11)₂

6.       Moreover, we can write 125.75₁₀ as (1111101.11)₂

However, of you take 0.9₁₀, notice that the multiplication and recording goes on and on without the fraction ever becoming zero.

0.90 x 2 = 1.80	1
0.80 x 2 = 1.60	1
0.60 x 2 = 1.20	1
0.20 x 2 = 0.40	0
0.40 x 2 = 0.80	0
0.80 x 2 = 1.60	1
0.60 x 2 = 1.20	1
0.20 x 2 = 0.40	0
0.40 x 2 = 0.80	0
0.80 x 2 = 1.60	1
0.60 x 2 = 1.20	1
0.20 x 2 = 0.40	0

… and on and on and on…. D;

For this occasion, we can use this alternative method:

1.       Multiply 0.9 by 2⁷ -> 0.9 x 128 = 115.2 which is in decimal

2.       Round off 115.2, giving you 115

3.       Convert 115 to binary, which gives you 1110011

4.       At the end of this, you’ll get (0.90)₁₀ = (0.1110011)₂,
which you will notice is the same as what you got above, in a much simpler method.

Converting Decimal to Octal

Use the same process of converting decimal to binary, but instead of  dividing by 2, divide by 8.

8	125	remainder
8	15	5
	1	7

ð  (125)₁₀ = (175)₈

Converting Decimal Fractions to Octal Fractions

Take 0.75 for example again ;}

however, instead of multiplying by 2, we multiply by 8

0.75 x 8 = 6.00

6

Therefore 0.75₁₀ would be 0.6₈

Converting Decimal to Hexadecimal

Use the same process of converting, but instead of 2 or 8, we divide the value by 16 instead.

16	125	remainder
16	7	13
	0	7

Since the 13^th unit in hexadecimal is D,

(125)₁₀ in hexadecimal would be (07D)₁₆

Converting Decimal Fractions to Hexadecimal fractions

The same as how you convert decimal to hexadecimal, you multiply by 16

0.75 x 16 = 12

12

Since the 12^th unit is C,

(0.75)₁₀ would be (0.C)₁₆ in hexadecimal.

-Xin Lin B031210345