Floating Point Binary Numbers (OCR A Level Computer Science)

Floating Point Binary

What is Floating Point Binary?

• Floating point binary addresses the limitations of fixed-point binary in representing a wide array of real numbers

• It allows for both fractional and whole-number components

• It accommodates extremely large and small numbers by adjusting the floating point

• It optimises storage and computational resources for most applications

Representation of floating point

The appearance of floating-point binary is mostly the same except for the presence the decimal point

An example positive floating point number

• In the example above, an 8-bit number can represent a whole number and fractional elements

• The point is always placed between the whole and fractional values

• The consequence of floating point binary is a significantly reduced maximum value

• The benefit of floating point binary is increased precision

Representation of negative floating point

• Negative numbers can also be represented in floating point form using two's Complement

• The MSB is used to represent the negative offset of the number, and the bits that follow it are used to count upwards

• The fractional values are then added to the whole number

Converting Denary to Floating Point

Denary to floating point binary

Example: Convert 6.75 to floating point binary

Step 1: Represent the number in fixed point binary. 

Step 2: Move the decimal point.

Step 3: Calculate the exponent 

The decimal point has moved three places to the left and therefore has an exponent value of three. 

Step 4: Calculate the final answer:

Mantissa: 011011

Exponent: 011

Converting Floating Point to Denary

Binary floating point to denary

Example: Convert this floating point number to denary:

• Mantissa - 01100

• Exponent - 011

Step 1: Write out the binary number. 

Step 2: Work out the exponent value.

The exponent value is 3. 

Step 3: Move the decimal point three places to the right. 

Step 4: Calculate the final answer: 6

Normalising Floating Point Binary

A floating point number is said to be normalised when it starts with 01 or 10. 

Why normalise?

• Ensures a consistent format for floating point representation

• Makes arithmetic and comparisons more straightforward

Steps to Normalise a Floating Point Number

1. Shift the decimal point left or right until it starts with a 01 or 10

2. Adjust the exponent value accordingly as you move the decimal point

   • Moving the point to the left increases the exponent and vice versa.

Example

Before normalisation:

• Mantissa = 0.0011

• Exponent = 0010 (2)

After normalisation:

• Mantissa = 0.1100 – Decimal point has moved 2 places to right so it starts with 01

• Exponent = 0000 (0) - Exponent has been reduced by 2