Unsigned binary and binary arithmetic

Overview

Recall questions on the range of an n-bit unsigned number, converting in both directions, adding and multiplying in binary, and recognising overflow.

What you need to know

Answer each one before you reveal it.

Questions from this subtopic

Each one with the model answer underneath.

1. 1

   What is the difference between unsigned binary and signed binary?

   Model answer

   In unsigned binary every bit contributes a positive place value, so only zero and positive numbers can be represented. In signed binary one pattern element — in practice the most significant bit — is used to indicate the sign, so negative numbers can be represented as well, but the largest positive value that fits in the same number of bits is smaller.

2. 2

   State the minimum and maximum values that can be held in unsigned binary in n bits.

   Model answer

   The minimum is 0, represented by all zeros, and the maximum is 2ⁿ − 1, represented by all ones.

3. 3

   Why is the largest unsigned value 2ⁿ − 1 rather than 2ⁿ?

   Model answer

   n bits give 2ⁿ different patterns, but one of them, all zeros, is used for the value 0. Counting starts at 0, so the last of the 2ⁿ patterns represents 2ⁿ − 1.

4. 4

   What is the range of an unsigned 8-bit number, and of an unsigned 16-bit number?

   Model answer

   8 bits give 0 to 2⁸ − 1 = 255. 16 bits give 0 to 2¹⁶ − 1 = 65 535.

5. 5

   Convert the unsigned binary number 10110101₂ to decimal.

   Model answer

   181₁₀. The columns holding a 1 are 128, 32, 16, 4 and 1, and 128 + 32 + 16 + 4 + 1 = 181.

6. 6

   Convert 181₁₀ to an 8-bit unsigned binary number.

   Model answer

   10110101₂. Work down the place values: 181 − 128 = 53, 53 − 32 = 21, 21 − 16 = 5, 5 − 4 = 1, 1 − 1 = 0, so there is a 1 in the 128, 32, 16, 4 and 1 columns and a 0 in the 64, 8 and 2 columns.

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