1. Binary Arithmetic

2. Binary Arithmetic We will see: How numbers are stored in a computer
The consequences of using fixed-length arithmetic
How to convert to and from binary
How to represent negative numbers
How to represent fractional numbers
Fixed-point
Floating-point

3. Why Binary Arithmetic? Early computers often used decimal arithmetic
Binary arithmetic is the most robust
Only need to distinguish between two different voltage levels
A decimal computer would need to distinguish ten different levels
Would need a much lower noise level to operate reliably – expensive
But could pack far more into a given word length
Another possibility is an analogue computer
But the presence of noise means low accuracy

4. 8 bits, 1 byte, 256 values
3 bits
8 combinations
4 bits
16 combinations
Each bit we add increases the number of possible combinations by 2
With n bits, we have 2n combinations
Hence 8 bits (1 byte) gives 28 = 256 different combinations

5. How many bits in a date?
Using binary values means we must divide our world into powers of 2
We usually pick the nearest power of 2
For the date format “DD/MM/YYYY”
29 to 31 days in a month: 25 = 32
12 months in a year: 24 = 16
9999 possible years: 214 = 16384
So we would need =23 bits in total to represent a date like this
This leaves many illegal bit combinations
E.g. “32/16/9999”

6. Fixed-length Arithmetic
We are used to working with numbers of any length
Computers have to work with fixed-length numbers
This causes some interesting problems
Suppose we only have 3 decimal digits
000, 001, 002, … 999
Cannot represent the following integers:
Negative numbers (no sign)
Fractional numbers (no decimal point)
Numbers larger than 999 (not enough digits)

7. Problems with Fixed-Length Arithmetic
Fixed-length arithmetic is not closed
For example, using 3 decimal digits:
= 1200 (too large)
= -2 (negative)
050 x 050 = 2500 (too large)
007 / 002 = 3.5 (not an integer)
We can divide the problems into three main classes
Overflow (too large)
Underflow (too small)
Not a member of the set

8. Binary-to-Decimal
000
0 = 0x22 + 0x21 + 0x20
Binary
Decimal
001
1 = 0x22 + 0x21 + 1x20
010
2 = 0x22 + 1x21 + 0x20
011
3 = 0x22 + 1x21 + 1x20
100
4 = 1x22 + 0x21 + 0x20
101
5 = 1x22 + 0x21 + 1x20
110
6 = 1x22 + 1x21 + 0x20
111
7 = 1x22 + 1x21 + 1x20
By convention, the right-most bit is the least-significant
Each subsequent bit is worth twice the one before
We can build this idea into an algorithm

9. Decimal-to-Binary
This involves the successive division of the decimal number by 2
The remainder at each stage is the next digit of the binary expansion
E.g. Converting 58 into binary
Divide 58 by 2 = 29 remainder 0
Divide 29 by 2 = 14 remainder 1
Divide 14 by 2 = 7 remainder 0
Divide 7 by 2 = 3 remainder 1
Divide 3 by 2 = 1 remainder 1
Divide 1 by 2 = 0 remainder 1
So 58 is in binary

10. Representations of Negative Integers
There are four main ways of representing negative m-bit binary numbers
Sign and magnitude
1’s Complement
2’s Complement
Excess 2m-1
All of them have problems
But all of them work well with binary addition and subtraction

11. Sign and Magnitude
We use one bit for the sign, and the rest for the size of the number
E.g. Using an 8-bit binary representation:
58 is
-58 is (using sign-magnitude)
The range is therefore –127 to 127
Problem:
= zero
= -zero

12. 1’s Complement Also has a sign bit
When negating a number, we simply flip every 0 to a 1 and visa versa
E.g. Using an 8-bit binary representation:
58 is
-58 is (using 1’s complement)
The range is again –127 to 127
We also have two representations for zero
= zero
= -zero

13. 2’s Complement
Like 1’s Complement, but to negate a number we negate all the bits and add 1 to the result
E.g. Using an 8-bit binary representation:
58 is
-58 is (using 2’s complement)
This gives us a range of –128 to 127
Only one representation of zero
But we still have a problem:
–(-128) = -( ) = = -128
InnerInt

14. Excess 2m-1 We store each number as its sum plus 2m-1
E.g. Using an 8-bit binary representation, this is “excess 128”
58 is represented as 186, or
-58 is represented as 70, or
This is equivalent to 2’s complement with the sign-bit reversed
The range is again –128 to 127
Still unequal

15. Fixed-Point Arithmetic
It is useful to be able to represent real numbers, e.g. 1.23, 3.141, …
One way we can do this is to reserve a certain number of digits to be the fractional part:
E.g = 10.75
(“.1100” = = = 0.75)
Note that we don’t need any special hardware to deal with this
We just need to keep track of the decimal point
Given m bits before and n bits after the decimal point
m determines the range
n determines the precision

16. Problems with Fixed-point Arithmetic
Fixed-point arithmetic is useful, but limited
No good for extremely large or small numbers
The mass of the Sun is 2x1033 grams
The mass of an electron is 9x10-28 grams
To use both these amounts in a calculation we would need over 60 decimal digits, most of which would be irrelevant
We need floating-point arithmetic

17. Floating Point Arithmetic
We borrow the familiar scientific notation to represent numbers:
3.14 = x 101
= 0.5 x 10-4
We use powers of 2 instead of 10:
N = mantissa x 2exponent (-1 < mantissa < 1)
Range depends on exponent
Precision depends on mantissa
This allows a huge range of numbers to be covered
With some loss in accuracy
IEEE floating-point standard 754:
Single precision: 1 sign bit, 8 exponent, 23 mantissa
Double precision: 1 sign bit, 11 exponent, 52 mantissa
Reserved values for infinity and NAN (not a number)
Inner Float

18. Problems with Floating-point Arithmetic
Negative
Overflow
Positive
Expressible
Negative Numbers
Positive Numbers
Underflow
The number line is divided into seven regions
Using floating point we can only access three of them
The range is not continuous, nor equally sampled
0.998x1099 and 0.999x1099 v.s x100 and 0.999x100
Cannot express some numbers at all, e.g x103 / 3
Need to round them to the nearest expressible number
For a fixed number of bits, we must trade off the range against the precision of the representation
Requires special hardware for fast performance

19. Summary Computers use a binary representation for numbers
They are bound by the constraints of fixed-length arithmetic
Representing negative numbers is something of a challenge
No system is ideal
We can represent real numbers using fixed-point or floating-point arithmetic
Again, neither system is ideal