Floating-Point Representation

This lesson covers how computers represent real numbers using floating-point representation. You need to understand the mantissa and exponent, normalisation, converting between floating-point and denary, and the trade-off between precision and range.

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Why Floating-Point?

Fixed-point binary has a fixed binary point position, limiting either range or precision. Floating-point allows the binary point to "float" — move to different positions — enabling representation of very large and very small numbers.

This is similar to scientific notation in denary:

• 3,400,000 = 3.4 x 10^6
• 0.00056 = 5.6 x 10^-4

In binary floating-point:

• The number is stored as a mantissa x 2^exponent

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Structure of a Floating-Point Number

A floating-point number consists of two parts:

Component	Purpose
Mantissa	Stores the significant digits (precision) of the number
Exponent	Stores the power of 2 (determines where the binary point sits)

Both the mantissa and exponent are stored in two's complement so that negative values can be represented.

Example Format: 8-bit Mantissa, 4-bit Exponent

[M M M M M M M M] [E E E E]
 ^                  ^
 mantissa            exponent
 (sign bit)          (sign bit, two's complement)

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Converting Floating-Point to Denary

Steps

1. Identify the mantissa and exponent bits
2. Convert the exponent from two's complement to denary
3. Place the binary point after the first bit of the mantissa
4. Shift the binary point right (positive exponent) or left (negative exponent) by the exponent value
5. Calculate the denary value from the resulting fixed-point binary

Worked Example

Mantissa: 01101000, Exponent: 0101 (i.e., +5)

Step 1: Write mantissa with binary point after bit 0 (the MSB): 0.1101000

Step 2: Exponent = 0101 = +5

Step 3: Shift binary point 5 places right: 0.1101000 -> 011010.00

Step 4: Calculate denary: 16 + 8 + 2 = 26.0

Worked Example: Negative Mantissa

Mantissa: 11010000, Exponent: 0011 (+3)

Step 1: Mantissa with binary point: 1.1010000

Step 2: Shift right by 3: 1101.0000

Step 3: This is a two's complement number. MSB is 1 so it is negative. Using place values: -8 + 4 + 1 = -3.0

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Converting Denary to Floating-Point

Steps

1. Convert the denary number to a fixed-point binary number
2. Determine how many places the binary point must be shifted to place it after the MSB
3. The number of shifts becomes the exponent (right shifts = positive, left shifts = negative)
4. The resulting binary (with point after the MSB) is the mantissa
5. Convert the exponent to two's complement binary

Worked Example: Convert 13.5 to Floating-Point (8-bit mantissa, 4-bit exponent)

Step 1: 13.5 in binary = 1101.1 Step 2: Write as 0.11011 x 2^4 (shifted the point 4 places right to normalise)

Wait — for positive numbers, the normalised form is 0.1xxxxx (MSB of mantissa is 0, next bit is 1).

So: 13.5 = 1101.1 = 0.110110 x 2^(4)... Actually, let us be precise:

Place binary point at the start: 0.11011 would require shifting 4 places right to get 1101.1

So mantissa = 01101100 (padded to 8 bits), exponent = 0100 (+4)

Result: Mantissa = 01101100, Exponent = 0100

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Normalisation

A floating-point number is normalised when the mantissa begins with:

• 0.1... for positive numbers
• 1.0... for negative numbers (two's complement)

Why Normalise?