Section 4.6

4 Data Representation

4.6 Floating Point Binary Numbers

With floating point numbers we store a mantissa and an exponent.

Example

7,800,000

becomes 0.78 × 10⁷.

The mantissa holds the digitsand the exponent determines the point where the fractional point goes.

Positive Mantissa

0110 1000 0000 0011 = 0.1101 × 2₁₀¹¹₂

Negative Mantissa

The mantissa is always written in Two's Complement form.

Negative Exponents

The exponent can also be negative as it is in Two's Complement form.

If the exponent's left hand bit is a 1 then, when converting the exponent to denary, the mantissa's bit pattern is shifted right.

The precision of the floating representation depends on how many digits can be stored in the mantissa.

In order to get the most accurate representation for a given number of digits in the mantissa the number is written with no leading zeros.

Example

0.0101 × 2₁₀⁰⁰⁰⁰ would be written as 0.101 × 2₁₀¹¹¹¹.

For a positive mantissa in normalised form the first two bits will always be 0.1.

Example

Normalise the number: 0 000110101 000010 (10-bit mantissa, 6-bit exponent)

Step 1 Put in the assumed binary point and convert the exponent to denary.	0	.	0	0	0	1	1	0	1	0	1
	Exponent = 000010₂ = 2₁₀
Step 2 Remembering that the overall value of the number must not change:
Shift the mantissa bit pattern three places to the left to make it start 0.1.	0	.	1	1	0	1	0	1	0	0	0
Subtract three from the exponent. Then convert it to binary.	2 - 3 = -1 -10₁₀ = 111111₂

The answer is: 0 110101000 111111

The first two bits of such numbers is always 1.0

To normalise a negative mantissa shift the mantissa left until the second bit is zero and subtract as necessary from the exponent.

Example

Normalise the number: 1 111100100 000011 (10-bit mantissa, 6-bit exponent)

Step 1 Put in the assumed binary point and convert the exponent to denary.	1	.	1	1	1	1	0	0	1	0	0
	Exponent = 000011₂ = 3₁₀
Step 2 Remembering that the overall value of the number must not change:
Shift the mantissa bit pattern four places to the left to make it start 1.0.	1	.	0	0	1	0	0	0	0	0	0
Subtract four from the exponent. Then convert it to binary.	3 - 4 = -1 -10₁₀ = 111111₂

With negative numbers with a magnitude greater than one it is better to normalise the equivalent positive number and carry out Two's Complement.

Next >