Binary Fraction Factory
Halves, quarters and eighths, made in binary.
Your mission
Welcome to the factory floor! Here, numbers are built from parts: whole units on the left of the binary point, and halves, quarters, eighths and sixteenths on the right. Find out what each part is worth, fill some production orders, then pass the quality check.
Explore
Tap the binary point, then switch on each of the four bits to its right.
Binary numbers can have a binary point, just like decimal numbers have a decimal point. Tap the point, then switch on each bit to its right to find out what it is worth.
0 of 5 parts inspected
Experiment
Fill all four production orders by setting the right bits.
Production orders
Make each of these on the factory line. They tick themselves off.
0 of 4 orders filled
Challenge
The display is broken! Build and read binary fractions using place values alone, in six orders.
Complete all 6 orders to continue.
Understand
Here is what the factory was showing you, in the words you need for the exam.
Place values on both sides of the point
Moving left, each place value doubles. Moving right, each place value halves. So the positions after the binary point are worth halves, quarters, eighths and sixteenths, or 2⁻¹, 2⁻², 2⁻³ and 2⁻⁴.
| Position | 2³ | 2² | 2¹ | 2⁰ | . | 2⁻¹ | 2⁻² | 2⁻³ | 2⁻⁴ |
|---|---|---|---|---|---|---|---|---|---|
| Place value | 8 | 4 | 2 | 1 | . | 1/2 | 1/4 | 1/8 | 1/16 |
| As a decimal | 8 | 4 | 2 | 1 | . | 0.5 | 0.25 | 0.125 | 0.0625 |
Binary to decimal
Add the place values of the 1 bits, on both sides of the point.
1011.101: the whole part is 8 + 2 + 1 = 11. The fraction is 1/2 + 1/8 = 0.5 + 0.125 = 0.625. So the total is 11.625.
Decimal to binary
Build the fraction from the biggest piece that still fits, just like in the factory. For 0.75: take 1/2 (0.25 left), then 1/4 (nothing left). So 0.75 = 0.11 in binary.
Some decimal fractions, like 0.1, can't be made exactly from halves, quarters and eighths, however many bits you use. Computers store the nearest value they can.
Words to know
- Binary point
- Separates the whole-number part from the fractional part of a binary number.
- Fractional place values
- 1/2, 1/4, 1/8, 1/16 and so on: negative powers of 2.
- Precision
- With 4 fraction bits, the smallest step is 1/16 = 0.0625.
Exam tip
The first position after the binary point is worth 1/2 (0.5), not 1/10! Write the fractional place values above the bits (0.5, 0.25, 0.125, 0.0625) before you add anything up, and show the whole and fractional parts separately.