1. Lecture 3: Binary values and number systems
Information Layer
Lecture 3: Binary values and number systems

2. Layers of Computing Systems
Communications
Applications
Operating Systems
Programming
Hardware
Information

3. Information Information Layer 01001010101010 numbers audio text Images
video

4. Binary values and number systems
Numbers and Computing
Positional notation
Binary, Octal and Hexadecimal
Arithmetic in other bases
Power of 2 Number Systems
Converting from base 10 to other bases
Binary values and computers

5. Numbers and Computing Computer executes numeric computations
All types of information that is stored on a computer is stored as numbers (0s and 1s)
Number – unit belonging to an abstract mathematical system and is a subject to specified laws of succession, addition and multiplication

6. Positional notation How many ones are there in 943?
The answer depends on base 
Base of a number system specifies the number of digits used in the system.
Digits always begin with 0 and continue through one less than the base
Eg. Base 10: has digits from 0 to 9

7. Base 10 (decimal)
Base 2 (binary)
Base 8 (octal)
Base 16 (hexadecimal) 1 2 10 3 11 4
100 5
101 6
110 7
111 8
1000 9
1001
1010 12 A
1011 13 B
1100 14 C
1101 15 D
1110 16 E
1111 17 F
11000 20

8. Positional notation
The rightmost digit represents its value multiplied by the base to the zeroth power, the digit to the left represents its value multiplied by the base to the first power
Number 943 demystified: 9 4 3
9*102
4*101
3*10 0
900 40
943

9. 9*x2+4*x1+3*x0
Formal notation
If a number in the base R number system has n digits, it is represented as follows, where di represents the digit in the ith position in the number
dn*Rn-1+dn-1*Rn d2*R+d1

10. dn*Rn-1+dn-1*Rn-2+...+d2*R+d1
Decompose number 2458 using the formula above
You can also use 9 4 3
9*102
4*101
3*10 0
900 40
943

11. Why?
Why would you need to represent values in base 2? 8? 13? 45???

12. binary to decimal
To convert a number from binary to base 10, just break the number into a sum of numbers from the above list.
Example: Convert to base 10
= = = 1910
= 16
+ 102 = 2
12 = 1
1910

13. Decimal to other bases While (the quotient is not zero)
Divide the decimal number by the new base
Make the remainder the next digit to the left of the answer
Replace the decimal number with the quotient

14. Decimal to other bases
Remainder
C B A
- ABC

15. Binary to octal
To convert from binary to octal, you start at the rightmost binary digit and mark the digits in groups of threes. Then you convert each group of three to its octal value.
E.g

16. Binary to hexadecimal
mark digits from right to left in groups of four.

17. Arithmetic in other bases
Basic addition in decimal (base 10)
0+1=1
1+1=2
2+1=3
9+1=10
Because there is no symbol for 10, we reuse same digits and rely on position. The rightmost digit reverts to 0 and there is a carry into next position to the left

18. Octal to decimal
(1336)8 = (1 x 83) + (3 x 82) + (3 x 81) + (6 x 80) = (1 x 512) + (3 x 64) + (3 x 8) + (6 x 1) = = 734 Thus (1336)8 = (734)10

19. Octal to binary
 look up each octal digit to obtain the equivalent group of three binary digits.
Octal  = 3 4 5
Binary =
011
100
101
= binary

20. Octal to hex
When converting from octal to hexadecimal, it is often easier to first convert the octal number into binary and then from binary into hexadecimal. For example, to convert 345 octal into hex:
Octal  = 3 4 5
Binary =
011
100
101
= binary

21. Octal to hex
Drop any leading zeros or pad with leading zeros to get groups of four binary digits (bits): Binary  = Then, look up the groups in a table to convert to hexadecimal digits.