1. Data Representation – numbers Binary conversion Hexadecimal Negative numbers Binary addition Binary shifts

2. Basic Binary
1 = switch closed / electricity on 0 = switch open / electricity off If you send 1 bit, how many different combinations can you send? 1 or 0 If you send 2 bits, how many different combinations can you sent? bits? 4 bits? 5 bits?

3. Terminology Bit 1 or 0 Nibble 4 bits Byte 2 nibbles / 8 bits KB
Kilobyte
1000/1024 bytes MB
Megabyte
1000/1024 KB GB
Gigabyte
1000/1024 MB TB
Terabyte
1000/1024 GB

4. Binary - Decimal
128 64 32 16 8 4 2 1 8 1
= 9
128 64 32 16 8 4 2 1
128 32 4 2
= 166

5. Have a go 1 2 3
128 64 32 16 8 4 2 1
128 64 32 16 8 4 2 1
128 64 32 16 8 4 2 1
Answers: 1 = 49, 2 = 154, 3 = 247, 4 = 158, 5 = 249, 6 = 741, 7 = 1543, 8 = 2405, 9 = 62423, 10 = 34582

6. Patterns
If the least significant bit (right most) is a 1, the number is odd
All 1s = the next number -1
e.g.
= 127
The smallest number in positive binary is always 0
The number of combinations is equal to the next number
= 128 different combinations
0 to 127
128 64 32 16 8 4 2 1
128 64 32 16 8 4 2 1

7. Decimal - Binary 23
23 – 16 = 7
7 – 4 = 3
128 64 32 16 8 4 2 1 98
98 – 64 = 34
34 – 32 = 2
128 64 32 16 8 4 2 1
242
242 – 128 = 114
114 – 64 = 50
50 – 32 = 18
18 – 16 = 2
128 64 32 16 8 4 2 1

8. Have a go 28 43 78
101
200
Answers: 1 = 11100, 2 = , 3 = , 4 = , 5 = ,
6 = , 7 = , 8 = , 9 = , 10 =

9. Hexadecimal Easier to remember than binary
1 nibble can be:
Decimal Hexadecimal
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
10 A
11 B
12 C
13 D
14 E
15 F
Hexadecimal
Easier to remember than binary
Quicker/easier to write than binary
Can be converted quickly to binary (and back)
Each nibble is converted into a single hexadecimal number

10. Hexadecimal - binary 3A 3 A 0011 1010 00111010 F20 F 2 0

11. Have a go, hex-bin 11 2A BB 6C A0 50F 9BD D5AA 1974 26ABEB
Answers: 1 = , 2 = , 3 = , 4 = , 5 = , 6 = ,
7 = , 8 = , 9 = , 10 =

12. Binary - Hexadecimal E
F5B

13. Have a go, binary-hex
Answers: 1 = 6A, 2 = BF, 3 = 80, 4 = 5F, 5 = E6B, 6 = 2AC, 7 = F0F, 8 = 33F, 9 = 5E2B, 10 = FF16

14. Hexadecimal - Decimal Convert to binary and then to decimal… Or… 161
160 3 A
3A =
Convert to binary and then to decimal…
Or…
(3 * 16) + (10 * 1) = 58
162
161
160
162
161
160 1 D 3
1D3 =
256 16 1
(1 * 16 * 16 ) + (13 * 16) + (3 * 1) = 467

15. Have a go, hex-dec 11 2A BB 6C A0 50F 9BD D5AA 1974 26ABEB
Answers: 1 = 17, 2 = 42, 3 = 187, 4 = 108, 5 = 160, 6 = 1295, 7 = 2493, 8 = 54698, 9 = 6516, 10 =

16. Decimal – Hexadecimal 78 = 4E 199 = C7
299
= 12B
256 16 1 4 14
Convert the binary and then hexa Or
256 16 1 12 7
162
161
160
256 16 1 2 11

17. Have a go – dec-hex 22 59
100
189
231
257
1056
2000
3578
32444
Answers: 1 = 16, 2 = 3B, 3 = 64, 4 = BD, 5 = E7, 6 = 101, 7 = 420, 8 = 7D0, 9 = DFA, 10 = 7EBC

18. Binary Addition What is 1 + 1? What is 0 + 0? What is 1 + 1 + 1?
What is 1 + 1? 1
What is 0 + 1? 1
What is ? 1

19. Binary addition – 4 basic rules1
0 + 0 = = = 0 carry = 1 carry 1 1

20. 1 1
Overflow = the result of the addition is too large to fit in 8 bits. A 9th bit is needed to store the result.
(1)

21. Have a go, binary addition1 1 1
(1) 1
(1) 1
If adding 4 1s, the result is binary , put a 0 in the box carry the 1 across two columns to the left 1
(1)
(1)

22. Negative numbers – Sign and Magnitude
The MSB bit (left most) gives the sign (it does not have a numeric value). 1 = negative, 0 = positive 0011 = = = = -10 Work out the binary – add a 0 or a 1 at the front

23. Have a go, dec – sign and mag22
-33 59
-100
105
-106
-221
398
-512
-1069
Answers: 1 = , 2 = , 3 = , 4 = , 5 = , 6 = , 7 = , 8 = , 9 = , 10 =

24. Negative numbers – Two’s complement Dec – two’s
Convert the number to binary as normal. Make sure there is a 0 in the MSB.
If the decimal number is positive, you already have the answer.
If it is negative, you need to flip it:
Method 2:
Right to left write down all the bits up to and including the first 1.
Flip the rest
Keep same:
Flip the rest:
Method 1: 1. Flip all the bits 2. Add Flip: Add 1:

25. Dec-two’s examples
59 Convert to binary (with a 0 at the front): is positive, so the answer is Convert to binary (with a 0 at the front): Flip all the bits: Add 1:

26. Have a go, Dec – two’s 22 -33 59 -100 105 -106 -221 398 -512 -1069
Answers: 1 = , 2 = , 3 = , 4 = , 5 = , 6 = , 7 = , 8 = , 9 = , 10 =

27. Two’s complement, two’s-dec
Check the first bit. If it is a 0 – convert it to decimal as normal. If it is a 1, you need to flip it, then work it out as normal:
Method 1:
1. Flip all the bits
2. Add 1
Flip:
Add 1:
Method 2:
Right to left write down all the bits up to and including the first 1.
Flip the rest
Keep same:
Flip the rest:

28. Two’s-dec examples
First bit = 0 so convert as normal = First bit = 1 so flip and add 1 Flip: Add 1: Convert: -108

29. Have a go, Two’s – dec
Answers: 1 = 53, 2 = -70, 3 = -64, 4 = 70, 5 = 85, 6 = -86, 7 = -166, 8 = -114, 9 = 3413, 10 = -5419

30. Binary Shifts
Move binary numbers a set number of places to the left, or the right
Logical shift – spaces are filled in with 0s
Arithmetic shift – when shifting left the spaces are filled with 0s, when shifting right they are filled with the MSB

31. Logical 1
Left shift 2 spaces 1 1
Right shift 2 spaces 1

32. Arithmetic 1
Left shift 2 spaces 1 1
Right shift 2 spaces 1

33. What do they do? Each left shift (log/ari) multiplies the number by 2
Each logical right shift divides the number by 2
Each arithmetic right shift divides the number by 2 but keeps the sign (positive/negative for two’s complement)
(Arithmetic left shift will keep the sign if bits are not removed)

34. Have a go, shifts Type Left/Right Num Places Binary 1 Logical Left 2
Arithmetic 3
Right 4 5 6 7 8 9 10
Answers