Fractions in Binary

1. Fractions in Binary

2. Fractions • Binary doesn’t just do integers (whole numbers) it can also be used for fractions though it isn’t as accurate as decimals in denary e.g. 136.75 could be expressed as 136 + 1/2 + 1/4 128 64 32 16 8 4 2 1 . 1/2 1/4 1/8 1/16 1 0 0 0 1 0 0 0 . 1 1 0 0 This is binary notation – the decimal point is in a fixed point

3. The problem • Clearly with a limited number of places after the decimal point it is far less accurate as only a limited range of decimals can be displayed • The number of extra places after the decimal point increase accuracy

4. Fixed point example • 1 byte = 8 bits If 4 bits were assigned after the decimal Place (this does not take up any bits) 8 4 2 1 . 1/2 1/4 1/8 1/16 0 1 1 1 0 1 0 1

5. Task • Look at the table on page 195 showing binary fractions • Do questions 1-3 on page 196

6. So how do you represent all of the other decimals in binary? • Floating point binary

7. Floating Point Binary • In decimal, you can use scientific notation for really big numbers • e.g. 1,2,0000,0000,0000 Would be 0.12 x 10 13 Exponent Which defines where to place the decimal point In this case move 13 places to the right Mantissa Which holds the digits

8. Floating Point Binary • In binary, firstly bits are shared between the mantissa and the exponent • In the example below, 2 bytes (16 bits) with 10 bits for the mantissa and 6 for the exponent Mantissa Exponent 0110100000 000011 = 0.1101 x 2 Sign bit 3 This is a sign bit – if positive i.e. 0 move the decimal point right, if negative i.e. 1 move the decimal point left There is always an imaginary decimal here

9. Floating Point Binary Mantissa Exponent 0110100000 000011 = 0.1101 x 2 = 110.1 = 6.5 in denary 3

10. Task • Do Q4 on page196 Rules: • Place the point between the sign bit and the first digit of the mantissa • Convert the exponent to its decimal form (positive or negative) • Move the point right if the exponent is positive or left if negative • Convert the resulting binary number to denary

11. If the mantissa is negative Mantissa Exponent 1110100000 000011 1.1101 would become 1110.1 -8 4 2 1 . ½ 1 1 1 0 . 1 = -8+4+2+0.5 = -1.5 Negative sign bit Move three places right Negative leftmost bit

12. What if the mantissa and exponent are negative? Mantissa Exponent 1100000000 111110 1.1 -2 Move the point left and fill with 1’s as you do so: 1.111 -1+0.5+0.25+0.125 = -0.125

13. Task • Do question 5 on page 197

14. Normalisation • In order to get the most accurate representation possible with a given size of mantissa, no zeros should be put to the left of the most significant bit (not including the sign bit) • E.g. in decimal: 0.034568x10 would be normalised to 0.34568x10 A binary number in normalised form will have the first bit of the mantissa not including the sign bit as a 1 9 8

15. To normalise a number in binary 0 000110101 000010 • put in the binary point and convert the exponent to decimal: 0.000110101 Exponent 2 • Move the number to the correct position and count the number of movements to the left to get the binary point before the first 1 0.110101000 required 3 left places • Take 3 away from the exponent of 2: 2-3 = -1 • Convert -1 back to binary 0 110101000 11111

16. To normalise a negative number in binary • With a negative number, the most significant bit not including the sign bit will be a 0 • Shift the number left until the first bit (not the sign bit) is a zero then adjust the exponent

17. To normalise a negative number in binary 1 111100100 000011 • put in the binary point and convert the exponent to decimal: 1.111100100 Exponent 3 • Move the number to the correct position and count the number of movements to the left to get the binary point before the first 0 1.001000000 required 4 left places • Take 4 away from the exponent of 3: 4-3 = -1 • Convert -1 back to binary 1.001000000 11111

18. Task • Do question 6 and 7 on page 198 • Read the hint on page 198 • Do exercises on page 199