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2.2 : Fractional Binary

Fractional Binary works just like the rest of binary. However, when we use powers, we're using negative powers. It's much easier to demonstrate below than to explain. We'll set up our table the exact same as we did before with an 8 bit number. However, this time, we're going to add a decimal point. This means we will have 4 integer bits and 4 fractional bits. Let's consider the number 1101.10112.

Binary to Base-10

Bit 1 1 0 1 . 1 0 1 1
Power 23 22 21 20 . 2-1 2-2 2-3 2-4
Fractional Value 8 4 0 1 . 1/2 0 1/8 1/16
Decimal Value 8 4 0 1 . 0.5 0 0.125 0.0625

I broke the Value row down into two other rows. Fractional Value and Decimal Value. The fractional value is the Power multiplied by the Bit. The decimal value is the same thing in decimal form. It's important to see the fraction form because it's much easier to see the powers of two. The decimal form makes it easier to add. So the value is 8 + 4 + 1 + 0.5 + 0.125 + 0.0625. 1101.10112 = 13.687510.

Base-10 to Fractional Binary

Converting the bits before the Radix (decimal point) is exactly the same as before with the alternating modulus/division method. Converting the bits after the radix is very different however. First you double the value, then if it greater than 1, you append a bit that is a 1. If the bit is greater than 1, you then subtract 1. Otherwise you append a 0. The loop continues until 0 is reached. Follow below through the method. We'll convert 0.210 to fractional binary.

Double Value: 0.2 2 = 0.4
Check Quantity: 0.4 < 1 : Append 0
Double Value: 0.4 2 = 0.8
Check Quantity: 0.8 < 1 : Append 0
Double Value: 0.8 2 = 1.6
Check Quantity: 1.6 > 1 : Append 1
Subtract 1: 1.6 - 1 = 0.6
Double Value: 0.6 2 = 1.2
Check Quantity: 1.2 > 1 : Append 1
Subtract 1: 1.2 - 1 = 0.2
Double Value: 0.2 2 = 0.4
Check Quantity: 0.4 < 1 : Append 0
Double Value: 0.4 2 = 0.8
Check Quantity: 0.8 < 1 : Append 0

Of course this particular number is never ending. Exactly like 1/3 is 0.3333333... In this case 0.210 is equal to 0.00110011...2.

Practice Problems

Whats is the value of 1000.0011 in base-10?


What is the value 0.562510 in base-2?


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