1. Binary – Octal - Hexadecimal
Intro to Computer Math
Binary – Octal - Hexadecimal B
0x B 4 C

2. Binary: What is it??? Electronics: 0,1 is easy numbering system
Data is stored as 32 or 64 bits of 0s and 1s:
Registers, Memory, Disk, CDs, DVDs, ROM, Data Communications
Bit = 1 binary digit: 1 or 0
Boolean: True = 1, False = 0
ASCII: 8 bits
Number = 8 or 16 or 32 or 64 (or 128) bits

3. Octal – Hexadecimal: What is it?
1s & 0s are tedious:
Hexadecimal: 0x a 5 f 3 c e 1
Octal:
Octal and Hexadecimal are compatible; powers of 2; easy representations of binary
Numbering Systems:
Base 2: Binary
Base 8: Octal
Base 10: Decimal
Base 16: Hexadecimal

4. Why is it Useful?
Uses include: UNIX Permissions: 740 = Self: Read, Write, Execute; Group: Read; Others: Null Data Communications: Meaning of: Transmission: 7E a b3 fd fa 4c 5d da a2 7f 7e Routing tables: 4a c0/26 Data: Setting flags, clearing bits, efficient storage Engineering: Designing and working with memory Debugging: Core dumps, reverse engineering

5. Look at Base 10 - Decimal Observations 0 1 2 3 4 5 6 7 8 9
Decimal System
Observations
Modulo = = = = (1x103) + (2x102) + (3x101) + 4

6. Understanding Different Bases
Other Bases
10 = = = =
B 10 = // Base 2 B 11 = B 110 = B 111 = x10 = // Base 16 0x11 = x 111 = 16*

7. Look at Base 2 - Binary Observations
Binary System
Observations 1
10 11
Modulo 2 10 = 21 = = 22 = = 23 = = (1x23) + (0x22) + (1x21) + 1 = 1110

8. Look at Base 8 - Octal Observations 0 1 2 3 4 5 6 7
Octal System
Observations …
Modulo 8 10 = 81 = = 82 = = 83 = = (1x83) + (2x82) + (3x81) + 4 = 66810

9. Look at Base 16 - Hexadecimal
Hexadecimal System
Observations
a b c d e f
a 1b 1c 1d 1e 1f
a 2b 2c 2d 2e 2f …
f0 f1 f2 f3 f4 f5 f6 f7 f8 f9 fa fb fc fd fe ff
a 10b 10c 10d 10e 10f
Modulo = 161 = = 162 = = 163 = a3f16 = (1x163) + (ax162) + (3x161) + f =

10. Working with Base 2: Binary
Each binary digit is a double of the digit to its right:
Binary can be noted as: B1011 or as 10112
So converting binary to decimal: (~=NOT)
B1011 = 8 + ~ = 11
B = 32 + ~ ~ ~1 = = 42
What is B ?

11. Equations with Carries
Adding with Binary
Equations with Carries
Obvious Equations
B B B
1 + 1 = = = 1101

12. Equations with Carries
Adding with Binary
Equations with Carries
Obvious Equations
carry: B B B Sum:
1 + 1 = = = 1101

13. NOT Operation ~
Examples: NOT
NOT Truth Table
The opposite of 0 is 1 Not_flag() If flag = false flag = true else flag = false
NOT ~
Input
Output 1
~0 = 1 ~1 = 0
~1011=0100 ~0101=1010

14. AND Operation - &
Examples: AND
AND Truth Table
If (you are >=18 AND you are registered to vote) then You can vote If (a==b && a==c) print(“a = b = c”);
AND & 1
1 & 1 = 1 0 & 0 = 0
1 & 0 = 0 0 & 1 = 0
If either result is false, the result is false.
If either input is false, result is false

15. OR Operation - |
Examples: OR
OR Truth Table
If (you are born in US OR you pass Citizen exam) then You are US Citizen If (a==b || a==c) print(“a is b or c”);
OR | 1
1 | 1 = 1 0 | 0 = 0
1 | 0 = 1 0 | 1 = 1
If either is true, the result is true.
If either input is true, result is true

16. XOR Operation - ⊕
Examples: XOR
XOR Truth Table
If (a!=b ) print(“no match = 1”); else // match print(“match = 0”); Uses: Encryption, etc.
XOR ⊕ 1
1 ⊕ 1 = 0 0 ⊕ 0 = 0
1 ⊕ 0 = 1 0 ⊕ 1 = 1

17. Anding/Oring Longer Numbers
AND Operation
OR Operation
& &
| |

18. Why ANDs and ORs?
AND Or
Useful for turning off bits Clear a field Clear a flag
Useful for turning on bits Set a flag Set a value into a field
Sign
1 bit
Exponent
8 bits
Fraction
23 bits

19. Convert Binary to Hexadecimal (Base 16)
Binary: B Step 1: Separate into 4 bits from the right: Binary: B Step 2: Now convert to Base 16: Hexadecimal: 0x193A77 Convert Base 16 to Binary: 0x193A77= B Easy to remember patterns, binary -> decimal B1010 = 0xA = 10 B1111 = 0xF = 15 B1100 = 0xC = 12

20. Converting Base 16 -> Base 10
Method 1: Convert to Binary, then Decimal:
Method 2: Use division remainders:
0x1af = = =
= 43110
0x456 = = = =
 Convert from base 10 to base N
(Example base 2):
Number / 2 ->remainder=>digit0
-> quotient / 2 -> remainder =>digit1
-> quotient / 2 -> remainder =>digit2

21. Converting Base 16 -> Base 10
Example Method 2:
Method 2: Use division remainders:
Example 1: Convert 3610 into binary: Quotient/2 ->Remainder 36/2 ->0 18/2 ->0 9/2 ->1 4/2 ->0 2/2 ->0 1/2 -> =
 Convert from base 10 to base N
Number / n ->remainder=>digit0
-> quotient / n -> remainder =>digit1
-> quotient / n -> remainder =>digit2
Example 2: Convert 3610 into base 16:
36/16 ->4
2/16 ->2
3610 =

22. Forming a negative Number
1's Complement
2's Complement
0= =1111? 1= =1110 2= =1101 3= =1100 4= =1011 Total: +7 -7
0=0000 1= =1111 2= =1110 3= =1101 4= =1100 Total: +7 -8

23. 2s Comp
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
0000

24. Signed & Unsigned Numbers
Binary
Signed
Unsigned 1 2
+126
+127
-128
+128
-127
+129
-126
+130 -2
+254 -1
+255
Assumes 1-byte storage

25. Signed & Unsigned Integers
Use when numbers may be negative
To create negative numbers, the high-order (top) bit is the signed bit.
0=Positive Number
1=Negative Number
Unsigned integers
Use when all numbers are POSITIVE.
No overflow to negative numbers are possible then
Accuracy - Security:
Do you want positive only – or positive & negative?
Signed: Incrementing goes from large positive to large negative.

26. Converting to Decimal: Powers of Two
The sign bit (bit 7) indicates both sign and value:
If top N bit is ‘0’, sign & all values are positive: top set value: 2N
If top N bit is ‘1’, sign is negative: -2N
Remaining bits are calculated as positive values:
= = = -86
= = = 85

27. Changing Signs: Two’s Compliment
A positive number may be made negative and vice versa using this technique Method: Take the inverse of the original number and add 1. Original: = = -85 invert: add 1: sum: = = 85

28. Changing Signs: Two’s Compliment
Why does this work? Identity + Inverse = -1 Number: Inverse: Sum: = -1 If x + x’ = -1 Then x + (x’ + 1) = 0 And x’ + 1 = -x

29. Shifting Bits
Shift Left
Shift Right
E.g.: 0xa9 = Shift left 1: Shift left 1: Shift left 2: New hexadecimal value:
E.g.: 0xa9 = Shift right 1: Shift right 1: Shift right 2: New hexadecimal value:

30. Shifting is useful
Move bits into position Extract a field Example: x 2101 Sign = 0 (positive) Exponent = B101 = 5 Shift and Or all fields together to get float value
Sign
1 bit
Exponent
8 bits
Fraction
23 bits

31. Short Cuts: Binary <-> Decimal
B1010 = 0xA =1010 B1011 = B1010+1=1110 B1111 = 0xF = 1510 B1110 = B1111-1
B1100 = 0xC =1210 B1101 = B

32. Conclusion
Binary – Octal - Hexadecimal
Computers operate in binary It is important to ‘speak’ binary and hexadecimal Base 2 & 16 will be used in a number of other courses