Decimal to Binary System

Conversion steps:

1. Divide the number by 2.
2. Get the integer quotient and remainder.
3. Divide the quotient again by 2.
4. Repeat the steps until the quotient is equal to 0.
5. Remainder of each step will give the binary digits.

Division by 2	Quotient	Remainder
13/2	6	1
6/2	3	0
3/2	1	1
1/2	0	1

Example: Convert 13₁₀ to binary

Therefore

13₁₀ = 1101₂

Division by 2	Quotient	Remainder
174/2	87	0
87/2	43	1
43/2	21	1
21/2	10	1
10/2	5	0
5/2	2	1
2/2	1	0
1/2	0	1

Example: Convert 174₁₀ to binary:

Therefore

174₁₀ = 10101110₂

Fractional Decimal to Binary System

Conversion steps:

1. Multiply the number by 2.
2. Get the product and the integer part.
3. Multiply the fractional part of the product by 2.
4. Repeat the steps until the fractional part is equal to 0.
5. Integer part of each step will give the binary digits.

Fraction	Product (Fraction x2)	Integer Part
0.125	0.250	0
0.250	0.500	0
0.500	1.0	1
0

Example: Convert 0.125 into Binary

Therefore,

(0.125)₁₀ = (0.001)₂

Binary to Decimal System

Conversion steps:

N = b_n qⁿ… b₃ q³ + b₂ q² + b₁ q¹ + b₀ q⁰ +….

Where b_x = Binary digits (bits: 0,1)              q^x = 2⁰,2¹,2²…

Example: 101₂ = (1 * 2²) + (0 * 2¹) + (1 * 2⁰) = 5

101100101₂ =1*2⁸+0*2⁷+1*2⁶+1*2⁵+0*2⁴+0*2³+1*2²+0*2¹+1*2⁰= 357

Fractional Binary to Decimal System

Above formula can be expanded into
N= b_-1 q^-1 + b_-2 q^-2 +….

Example:

0.101₂ = (1*1/2) + (0*1/2²) + (1*1/2³)

0.101₂ = 1*0.5 + 0*0.25 + 1*0.125

0.101₂ = 0.625

Therefore 101.101₂ =5.625

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