A2 · Practice questions · Floating-point binary representation

Mantissa times two to the power.

Six original Cambridge-style questions on floating-point format and conversions, every answer checked, with mark points.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

Keep these straight

Exponent first.

Value = mantissa × 2^exponent; both in two's complement.
Mantissa place values: −1 . ½ ¼ 1/8 ...; the point follows the first bit.
To denary: exponent, mantissa, multiply (or move the point); show each step.
From denary: binary, slide the point to 0.1, count places for the exponent; negate the mantissa for negatives and check it starts 10.

Recall

[2 marks]

Describe how a real number is represented in binary floating-point form.

As a mantissa and an exponent, both stored in two's complement; value = mantissa × 2^exponent. ✓
The mantissa holds the significant digits (with the binary point after its first bit); the exponent gives the number of places the binary point moves. ✓

Convert

[3 marks]

A floating-point number uses an 8-bit mantissa and a 4-bit exponent, both in two's complement. Convert 10110000 0010 to denary. Show your working.

Exponent 0010 = 2. ✓
Mantissa = -1 + 0.25 + 0.125 = -0.625. ✓
-0.625 × 2² = -2.5. ✓

Convert

[3 marks]

Using the same format, convert 01001000 1101 to denary.

Exponent 1101 = -8 + 4 + 1 = -3. ✓
Mantissa 0.1001 = 0.5 + 0.0625 = 0.5625. ✓
0.5625 × 2^-3 = 0.0703125. ✓

Convert

[3 marks]

Show the denary number 7.75 as a normalised floating-point number with an 8-bit mantissa and a 4-bit exponent.

7.75 = 111.11. ✓
= 0.11111 × 2³. ✓
Mantissa 01111100, exponent 0011. ✓

Convert

[4 marks]

Show the denary number -0.375 as a normalised floating-point number with an 8-bit mantissa and a 4-bit exponent.

0.375 = 0.011 = 0.11 × 2^-1, so the positive mantissa is 01100000. ✓
Two's complement of the mantissa: 10100000. ✓
Exponent -1 = 1111. ✓
Answer 10100000 1111 (check: -0.75 × 0.5 = -0.375). ✓

Convert

[3 marks]

A different system uses a 10-bit mantissa and a 6-bit exponent, both in two's complement. Convert 0101100000 000100 to denary.

Exponent 000100 = 4. ✓
Mantissa 0.101100000 = 0.5 + 0.125 + 0.0625 = 0.6875. ✓
0.6875 × 2⁴ = 11 (or move the point: 01011.00000 = 11). ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.