C10 · Topic 2.1 · 40 minutes · DAT-1.A, DAT-1.B, DAT-1.C
Binary values and representation limits
By the end you can:
- Convert unsigned binary using place values.
- Calculate a fixed-width range.
- Explain overflow under an explicit convention.
Start here; no earlier topic is required.
- Bit
- A binary digit,0 or 1.
- Sampling
- Taking measurements of a continuous quantity at selected times.
Understand
Unsigned positional binary uses place values 1,2,4,8 and successive powers of 2. Add the places containing 1 to decode a value; leading zeros do not change it. With n bits there are 2ⁿpatterns and unsigned values 0 through 2ⁿ−1.
A fixed-width representation cannot encode every integer. When a result exceeds its range, behaviour depends on the stated system convention. Here overflow is reported rather than silently wrapping. One extra bit doubles the number of patterns; it does not merely add one more value or remove every possible overflow.
Worked example
Encode 45 in six bits, then test the endpoints.
- 45=32+8+4+1, so the pattern is 101101.
- Six bits give 64 patterns and a maximum of 63, encoded 111111.
- 64 requires 1000000(seven bits); the stated six-bit model reports overflow.
Explore each concept
Choose a concept, change its inputs and follow the explanation. Each example models the stated rule or scenario; completing it is practice rather than evidence of mastery.
- Analog versus digital
- Sample intervals and rounding
- Bits, bytes and conventions
- Fixed-width versus AP integer model
Change it · step through · explain
Change an input, predict what will happen, then advance one step. Read why the state changes, reset and experiment. This fixed teaching model does not save progress or award XP.
Ready · Make a prediction
- ExploresamplingWaiting
- Sampling interval1Waiting
- Stored precisionwholeWaiting
Ready to explore: Sampling, precision and bit interpretation
Why: Change an input, predict the first effect, then advance one step.
Try this: Change one input or choose another listed case. Predict the effect, reset and compare the states and governing rule.
Step through and experiment
Change it · step through · explain
Change an input, predict what will happen, then advance one step. Read why the state changes, reset and experiment. This fixed teaching model does not save progress or award XP.
Ready · Make a prediction
- Input 145Waiting
- Input 26Waiting
Ready to experiment
Why: Predict the first change, then advance one step.
Try this: Try 63 and 64, then change the width.
Check a misconception
Claim: Adding one bit adds one representable value.
Every old pattern can be prefixed by 0 or 1, doubling the number of patterns.
Guided practice
Decode 010110 and state why the leading 0 does not change it.
Work with a partner. Make a prediction, compare your reasoning, then identify the rule that supports your answer.
Independent transfer
A width changes from 5 to 6 bits. Give both unsigned maxima and the pattern-count change.
Exit check
Why must an overflow prediction name the convention?
Write your reasoning before comparing with a classmate. If your teacher assigned this task, submit it through My classes to save your work and receive feedback.
Check yourself
Your practice so far
Check an answer to see its rationale.