floating-point-binary-numbers

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

Mantissa and exponent

• In A Level Computer Science, the mantissa and exponent are the two main components of a floating point number:

  • Mantissa

    • The part of the number that holds the actual digits of the value

    • It represents the precision of the number but does not determine its scale

  • Exponent

    • Controls how far the binary point moves, effectively scaling the number up or down

    • A larger exponent moves the point to the right (making a larger number), while a smaller exponent moves it to the left (making a smaller number)

• For example, in standard scientific notation, the decimal number 3.14 × 10³ has:

  • Mantissa: 3.14 (the significant digits)

  • Exponent: 3 (which shifts the decimal point three places to the right)

• Floating point binary works similarly but in base-2

• Instead of multiplying by powers of 10, it multiplies by powers of 2 using a binary exponent

Representation of floating point

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

• 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

Decode a normalised floating point binary number

• In A Level Computer Science, you may be given a floating point number made up of two parts:

  • A 4-bit mantissa (in two’s complement)

  • A 4-bit exponent (also in two’s complement)

• For example:

  • 1101 1111

  • Mantissa = 1101

  • Exponent = 1111

Step 1: Convert the exponent to denary

• The exponent is stored in 4-bit two’s complement, so:

  • If the first bit is 0 → it’s a positive number

  • If the first bit is 1 → it’s negative

• Example:

Exponent: 1111 → two’s complement = -1

Step 2: Convert the mantissa to binary

• The mantissa is also in 4-bit two’s complement

• Convert it to a denary number.

• Then convert it to a binary value, assuming the binary point goes just after the first bit (because it’s normalised)

• Example:

Mantissa: 1101 → two’s complement = -3

Binary of 3 = 011

So -3 in normalised binary = -0.110 (we assume a leading 1 is implied)

Step 3: Shift the binary point

• Now shift the binary point by the exponent value.

  • If exponent is positive, move the point to the right

  • If exponent is negative, move it to the left

• Example:

Start with: -0.110

Exponent = -1

Shift the point 1 place left → -0.0110

Step 4: Convert to denary

• Now convert the final binary number into denary

-0.0110 =

0 × ½ = 0

1 × ¼ = 0.25

1 × ⅛ = 0.125

0 × 1/16 = 0

Final value = -0.25 - 0.125 = -0.375

• Answer: –0.375