Topic 1c — Binary Fractions

Understanding how 8‑bit strings can represent numbers with both integer and fractional parts using binary place value.

Learning Objectives

By the end of this topic, you should be able to:

This topic has one focused skill and one reading — that's intentional. It builds directly on Topic 1b and the concept is not complicated once you see it. Spend your time on the practice problems rather than looking for more content.

Learning Activities

To help you meet the learning objectives, we have prepared a combination of readings, activities, and videos.

Course Readings

These reading were designed to introduce the course topics to an audience of educators. They should be considered "required" and read in order.

Supplemental Readings

Some participants find it helpful to read about a topic from a source written for a slightly more technical audience. These supplemental readings cover similar material as the course readings but may not fully align with the course learning objectives. Use them as an optional complement to your study, not a substitute for the course readings.

Lesson Videos

These videos support the readings above and may present the material with some deeper connections and worked examples.

Checking for Understanding, Questions

Review the Learning Objectives from the top of this page. You will be asked to complete these types of problems on this week's competency demo. To help you check if you are on the right track with these concepts, consider the following questions. Try each one on your own before looking at the answer key. It is completely fine if you need to revisit the reading or video — that’s part of learning.

Binary Fraction → Decimal

  1. Convert the 8‑bit binary value 110110102 into a decimal value using 4 integer bits and 4 fractional bits.
  2. Convert the 8‑bit binary value 101111102 into a decimal value using 4 integer bits and 4 fractional bits.

Interpretation Questions

  1. Which fractional bits are "turned on" in 010011012?
  2. How many unique numbers can be stored using 8 bits if the first 4 are integer bits and the second 4 are fractional bits?

Checking for Understanding, Answers

You can begin be comparing your answers to the following answer key.

Show Answer Key

Binary Fraction → Decimal

  1. 1101 . 1010₂
    Integer part (1101₂): 1×8 + 1×4 + 0×2 + 1×1 = 13
    Fractional part (1010₂): 1×1/2 + 0×1/4 + 1×1/8 + 0×1/16 = 0.5 + 0 + 0.125 = 0.625
    Final value: 13.625
  2. 1011 . 11102
    Integer part (1011₂): 1×8 + 0×4 + 1×2 + 1×1 = 11
    Fractional part (1110₂): 1×1/2 + 1×1/4 + 1×1/8 + 0×1/16 = 0.5 + 0.25 + 0.125 = 0.875
    Final value: 11.875

Interpretation Questions

  1. 0100 1101₂
    Fractional bits are the last four: 1101
    "Turned on" positions (bits = 1): 1/2, 1/4, and 1/16
    (The only fractional place with a 0 value (and, thus, not included) is 1/8.)
  2. Integer bits: 4 bits → 16 possible integer values (0–15)
    Fraction bits: 4 bits → 16 possible fractional values (0–15)/16

    Total unique representable numbers: 16 × 16 = 256 unique values

    Or thinking about it another way: 8 bits produces 28 or 256 values. It doesn't matter how we interpret those values — there are still only 256 of them.

If you still have questions or concerns you can use the videos below to try and further your understanding.

Extend Your Learning

The following resources go a little deeper on topics we touched on but did not fully explore in the readings. These are entirely optional — none of this material appears on the Competency Demo — but each one is a natural "next question" from something covered this week.

  • Floating-point representation
    Our simplified 4‑bit integer / 4‑bit fraction encoding gives you the core idea, but real computers use a much more sophisticated system called floating-point. This Wikipedia article explains how the IEEE 754 floating-point standard works — the same format used by virtually every modern processor and programming language.
    Floating-Point Arithmetic — Wikipedia