1. The number systems of Computer Science
Binary & Hex
The number systems of
Computer Science

2. What you will learn Decimal Binary Hexadecimal Arithmetic
Converting between number bases
Counting
2’s Compliment

3. Number Systems Prehistory Unary, or marks: / /////// = 7
/////// + ////// = /////////////
Babylonian (Mesopotamia) around 2000 BC
Used 60 numerical symbols – sexadecimal. Used to measure time
5; 25, 30 = 5hr 25min 30 secs. Was positional based by had no concept of zero. 5,4560 = 5 x x 1 = 34510
Roman Numerals: (non positional, no concept of zero)
VII + V = VVII = XII
MCMXCIX = 1999
MCMXCVIIII = 1999
MCMLXXXXVIIII = 1999
MDCCCCLXXXXVIIII = 1999

4. Quote from Pierre Simon Laplace
“The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius. “

5. Arabic/Indian Numerals (Decimal)
Better, Arabic/Indian based Numerals:
7 + 5 = 12 = 1 x
345 is really
3 x x x 1
3 x x x 100
3 is the most significant symbol (carries the most weight)
5 is the least significant symbol (carries the least weight)
Digits (or symbols) allowed: 0-9
Base (or radix): 10

6. Try multiplication in (non-positional) Roman numerals(!):
XXXIII (33 in decimal)
XII (12 in decimal)
XXXIII
CCCXXX
CCCXXXXXXXXXIIIIII
CCCLXXXXVI
CCCXCVI = 396
The Babylonians wouldn’t have had this problem.

7. There are many ways to “represent” a number
Representation does not affect computation result
LIX + XXXIII = LXXXXII (Roman)
= (Decimal)
Representation affects difficulty of computing results
No concept of “Zero”
Computers need a representation that works with fast electronic circuits

8. What is the base ?
The decimal numbering system is also known as base 10. The values of the positions are calculated by taking 10 to some power.
Why is the base 10 for decimal numbers ?
Because we use 10 digits. The digits 0 through 9.

9. Main Memory
Original computers used holes/no hole to represent two state, 1/0.
Modern computers use transistor on/off translates to values 1/0
Requires use of Binary number system

10. Binary: positional numbers work great with
2-state devices
Digits (symbols) allowed: 0, 1
Binary Digits, or bits
Base (radix): 2
10012 is really
1 x x X X 20
910
is really ?
Computers usually multiply Arabic numerals by converting to binary, multiplying and converting back for display

11. Counting in Binary Binary 1 10 11 100 101 110 111 Decimal equivalent 11 10 11
100
101
110
111
Decimal equivalent 1 2 3 4 5 6 7

12. Decimal to Binary Divide decimal value by 2 until the value is 0 E.g.
47 /2 = 23 Remainder 1
23/2 = 11 Remainder 1
11/2 = 5 Remainder 1
5/2 = 2 Remainder 1
2/2 = 1 Remainder 0
1/2 = 0 Remainder 1
Read from the bottom up
Ans:

13. Binary to Decimal
X 20 = 1
X 21 = 0
0 X 22 = 0
20 = = X 23 = 0
21 = = X 24 = 0
22 = = X 25 = 0
23 = X 26 =

14. Binary to Decimal (alternative)
Converting the binary to 37
Multiply running total by two, add in next digit. Repeat for all digits.

15. Some buzz words Single Binary Digit =Bit 4 bits = nibble 8 bits = byte
16 bits = word
32 bits = longword
2 nibbles = 1 byte
2 bytes = 1 word
The word size depends on the processor used, above is shown the word size for the 68000

16. Addition of Binary Numbers
Examples:
Carry a one

17. Addition (large numbers)
Example showing larger numbers:

18. Working with large numbers
Humans can’t work well with binary numbers. We will make errors.
Shorthand for binary that’s easier for us to work with - Hexadecimal

19. Hexadecimal Binary 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 1 Hex 5 0 9 7
Written:

20. What is Hexadecimal really ?
Binary
Hex
A number expressed in base 16. It’s easy to convert binary to hex and hex to binary because 16 is 24.

21. Counting in Hex Binary Hex Binary Hex 0 0 0 0 0 1 0 0 0 8 A B C D E F

22. Decimal to Hex
Repeat dividing by 16 and noting the remainder until the quotient = 0
E.g d to Hex
54401/16 = 3400 Remainder 1
3400/16 = 212 Remainder 8
212/16 = 16 Remainder 4
13/16 = 0 Remainder 13
Remainders are expressed Hex numbers
Answer: $D481

23. Hex to Decimal Write out the 1st 16 numbers
Same as for binary to decimal
E.g. F9E116
1x160 = 1
Ex161 =14x16 = 224
9x =9x256 = 2304
Fx163 =15x4096= 61440
= 63969

24. Another Binary to Hex Conversion
Hex C F
7C3F16

25. Binary Coding Schemes
To represent characters we use Binary Coding Schemes
Two of the most popular are
ASCII (American Standard Code for Information Interchange
EBCDIC (Extended Binary Coded Decimal Interchange Code)
Here are some ASCII codes
CHARACTER ASCII
A – 41 in hex
B in hex
C in hex
D in hex
BELL – 7 in hex
ESC – 1B in hex

26. Binary Coded Decimal (BCD)
Each digit is converted individually to it’s binary equivalent
2634 =
3825 =
Disadvantages are that arithmetic is more complicated and storage is inefficent.
Used only in simple device where memory is not a problem e.g. Calculators, clocks, etc.

27. Negative Integers Most humans precede number with “-” (e.g., -2000)
Accountants, however, use parentheses: (2000)
Sign-magnitude
Example: in hex?
= 3 x e x x 160
-3E816

28. 2’s Compliment Used for negative binary numbers Procedure: E.g.
Invert all the bits: 0 becomes 1 and 1 becomes 0
Add 1
E.g.
To get –1
110 =
Invert all bits
Add
Ans: -110 =

29. 2’s Compliment
Using the TC notation, the Most significant indicates the sign. 1 means a negative number
With 8 bits, can represent signed numbers in the range:
With N bits, can represent signed numbers in the range:
(2N-1)-1 0 -(2N-1)
TC is widely used because it is easy to manipulate. The sign bit is treated the same as the others

30. 2s Compliment Clock Stepping clockwise increases the binary number
Stepping anti-clockwise decreases the binary number
Notice how when we add one to a negative number it’s values move closer to zero

31. Addition & Subtraction
Addition of TC’s numbers same as normal binary addition
Subtraction of TC’s numbers can use either
A-B or A + TC(B)
E.g.
01112 – = ???
Change second number to TC
0111
1101+
0100
Answer: – = 01002

32. Decimal fractions to binary
More info on p158 Clements.
Question: Convert (decimal) to a binary fraction
x 2 = yielding 0
0.625 x 2 = yielding 1
= yielding 0
0.5 x 2 = yielding 1
Ans:0.0101

33. Binary to Decimal fractions
More info on p158 Clements.

34. Summary Decimal Binary Hexadecimal Arithmetic
Converting between number bases
Counting
Coding schemes
2’s Compliment