Binary Real Numbers

Introduction

Computers must be able to represent real numbers (numbers w/ fractions)

Two different ways: Fixed-point Floating-point

NOTE: Everything in binary uses powers of two

Decimal Review

Digits to the right of the decimal point correspond to negative powers of 10

102 101 100 . 10-1 10-2

100 10 1 . 0.1 0.01

Binary Fractions

2-1 2-2 2-3 2-4 2-5 2-6

0.5 0.25 0.125 0.0625 0.03125 0.0015625

1/2 1/4 1/8 1/16 1/32 1/64

Fixed Point Notation

1. Multiply each 1 by the corresponding power of 2

2. Add up the resulting powers of 2

Example:

11.012 = 2 + 1 + ¼ = 3.2510

00111.0102 = 4 + 2 + 1 + ¼ = 7.2510

Floating-Point Notation

Floating-point notation is essentially the computer’s way of storing a number that has been normalized

3 different parts of any number: Mantissa: normalized number Exponent: power to which the base is raised Sign: of both mantissa and exponent

Decimal Example:12.5 = 0.125 x 102 normalized!

Normalization Steps

1. Beginning with a fixed point number

2. Normalize the number such that the radix point (decimal point) is all the way to the left (produces the mantissa)

3. Multiply the resulting number by the base raised to an exponent

Floating-Point ExampleWhat is 12.5 in floating-point representation?1. Convert 12.5 to binary fixed point

12.510 = 1100.12

2. Normalize the number by moving the radix point, producing the mantissa

1100.12 = 0.11001 * 24

3. Fill in the bits for each of the three parts of any real number:1. Sign (2 bits)2. Mantissa (# bits varies)3. Exponent (# bits varies)

4. NOTE: 2’s complement may be applied to the mantissa or the exponent if either are negative

Placing the Bits

Assume you have the following: 1 bit for the mantissa sign 8 bits for the mantissa 1 bit for the exponent sign 6 bits for the exponent

SM M M M M M M M M S E E E E E E E

Example:

0.11001 * 24 0 00011001 0 000100

Another Example

Convert -12.510 to binary:

1. Convert 12.5 to fixed point 01100.1

2. Normalize 0.11001 * 24

3. Convert exponent base to binary: 4 0100

4. 2s complement the mantissa by flipping bits and adding 1: 011001 100111

5. Final number 1 00111 0100

Upper & Lower Bounds Assume you have the following:

1 bit for the mantissa sign 8 bits for the mantissa 1 bit for the exponent sign 6 bits for the exponent

What is the upper bound for the floating-point number?

What is the lower bound for the floating-point number?

What happens if we convert a floating-point number to an integer?

Integers vs. Floating-point

integers: smaller range than floating-point all numbers within the range are 100% accurate

floating-point large range of numbers not all numbers within the range can be

represented accurately Example: 2.9999999999999 repeating

Possible Errors

truncation error round off errors using floating-point numbers

because not all real numbers can be represented accurately

overflow error attempting to represent a number that is greater

than the upper bound for the given number of bits underflow error

attempting to represent a number that is less than the lower bound for the given number of bits