Encode a Value and Check the Range

Supplied materials

• Paper and pencil; an adult may read aloud. No device, account or purchase is needed.

Learn the idea

Encoding reverses decoding: choose which place values sum to the desired number. For unsigned four-bit values the places are 8, 4, 2 and 1. Start with the largest place and record 1 if you can include it without exceeding the remaining value. Otherwise record 0. Keep all four positions.

Encode 10: include 8, leaving 2; skip 4; include 2, leaving 0; skip 1. The code is 1010. The largest code 1111 represents 8 + 4 + 2 + 1 = 15. A value of 16 cannot be represented in these four unsigned places; we reject it rather than silently dropping a digit.

1. Write columns place, included? and amount still needed.
2. Encode 10 using places 8, 4, 2, 1.
3. Decode your result to verify it adds back to 10.
4. Test the lower endpoint 0 and upper endpoint 15, then reject 16.

Check your reasoning

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Take it further

Change the width to five unsigned bits with places 16, 8, 4, 2, 1. Expected encoding of 16: 10000; largest value: 31.

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