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Lesson 5. Normalisation

Lesson Notes

Normalisation

There are two main reasons why we need to normalize floating point binary numbers:

1. To ensure maximum accuracy. When a floating point number is normalized, the mantissa (the part of the number to the right of the decimal point) is as large as possible without overflowing the number. This means that the number can be represented with the least number of bits, which in turn gives the greatest possible accuracy.
2. To ensure uniqueness. When a floating point number is normalized, each unique number has only one possible bit pattern to represent it. This is important for ensuring that floating point operations are performed correctly.

It is the process of moving the binary point of a floating point number to provide the maximum level of precision for a given number of bits.

• To do this for a positive binary number involves removing any leading zeros (0s).
• To do the same for a negative binary number involves removing and leading ones (1s).
• This means that a normalised floating point number must always start as either 0.1 or 1.0.

Normalise a Positive Floating Point Number

Example: 14.125 into a normalised two's complement floating point number with a 10 bit mantissa and a 6 bit exponent.

8 + 4 + 2 + ⅛(0.125) = 14.125

Positive two's complement numbers most significant bit must be 0. In order to maximise precision the next most significant bit must be 1.

Place the decimal point between the first 0 and 1. Move 4 places to the left.

The value of the exponent is 4 because it would have to be moved 4 places to the right to return to the original number.

The normalised two's complement floating point representation of 14.125 is below.

Normalise a Negative Floating Point Number

Example: -45.375 into a normalised two's complement floating point number with a 10 bit mantissa and a 6 bit exponent.

-64 + 16 + 2 + ½(0.5) + ⅛(0.125) = -45.375

In a negative two's complement format the most significant bit must be 1.

In order to maximise precision the next most significant bit must be 0.

We must place the decimal point between the first 1 and 0. In this case we need to move the binary point 6 places to the left.

The normalised two's complement floating point representation of -45.375 is below.

Normalise a Positive Floating Point Number - Negative Mantissa

Example: -0.46875 into a normalised two's complement floating point number with a 10 bit mantissa and a 6 bit exponent.

-16 + 8 + 4 + 2 + 1 + ½(0.5) + 1/32(0.03125) = -0.46875

The decimal point between the first 1 and 0. In a negative number leading 1s can be lost so in this case we need to move the binary point 1 place to the right and lose 5 leading 1s.

The binary point was moved 1 place to the right so the value of the exponent is -1 because it would have to be moved 1 place to the left to return to the original number.

Final number is below: