1. Number Systems & Binary
How to count

2. How do we represent numbers?
Early systems:
Actual count : ||||| = 5
Roman numbers : XI = 11
Hard to do math:
MCXXIII + CLXXIV = ????

3. Place Value Systems What we are used to
Digits that have different meaning based on place:
= 1 * * * * * 1
= 12059
ten thousands
thousands
hundreds
tens
ones 1 2 5 9

4. Decimal Numbers Base 10 : Each place increases by a power of 10 1 2 5
ten thousands
104
thousands
103
hundreds
102
tens
101
ones
100 1 2 5 9

5. Why Base 10? Only one thing special about base 10:
Other systems have been used:
Mayans : Base 20
Babylonians : Base 60

6. Other Bases Base determines the multiplier from one column to next
= 1 * * * * * 1 =
160-thousands
204
eight-thousands
203
four-hundreds
202
twentys
201
ones
200 1 2 5 9

7. Other Bases Base also determines how many digits needed:
Base 10 : 10 digits : 0-9
Base 2 : 2 digits : 0-1
Base 8 : 8 digits : 0-7

8. Base 2 Base two Only 1 and 0 Places are powers of 2 1
= 1 * * * * * * 1 =
= 45
thirty-twos 25
sixteens 24
eights 23
fours 22
twos 21
ones 20 1

9. Notation Subscript used to indicate base:
1012  101 in base 2 (numeric value of 5)
10110  101 in base 10 (numeric value of 101)

10. More digits = bigger numbers
4 digits we can represent values :
8 digits can represent 0-255
16 digits can represent
n digits can represent 0 to (2n-1) 8 4 2 1

11. Counting to ten in binary
Values for 1-10 (decimal) in binary: 1
0001 6
0110 2
0010 7
0111 3
0011 8
1000 4
0100 9
1001 5
0101 10
1010

12. Binary -> Decimal with Table
Make a table like this one
Write in the binary number
Add up all the numbers from the top that have a 1 under them:
= 101
128 64 32 16 8 4 2 1
128 64 32 16 8 4 2 1

13. Decimal -> Binary with Table
Make a table like this one
Say we are converting 83:
Look for largest number that is smaller than our value
128 64 32 16 8 4 2 1

14. Decimal -> Binary with Table
Make a table like this one
Say we are converting 83:
Look for largest number that is smaller than our value
We have 1 of those (0 of anything bigger)
Subtract that from our number 83 – 64 = still have 19 to make
128 64 32 16 8 4 2 1

15. Decimal -> Binary with Table
Make a table like this one
Say we are converting 83:
Look for largest number that is smaller than our value
We have 1 of those (0 of anything bigger)
Subtract that from our number 83 – 64 = still have 19 to make
Look for largest value smaller than that… Now have 19 – 16 = 3 left
128 64 32 16 8 4 2 1

16. Decimal -> Binary with Table
Make a table like this one
Say we are converting 83:
Look for largest number that is smaller than our value
We have 1 of those (0 of anything bigger)
Subtract that from our number 83 – 64 = still have 19 to make
Look for largest value smaller than that… Now have 19 – 16 = 3 left
Continue… need a 2, have one left…
128 64 32 16 8 4 2 1

17. Decimal -> Binary with Table
Make a table like this one
Say we are converting 83: means = 83
128 64 32 16 8 4 2 1

18. Conversion With Division/ Multiplication
Binary works in powers of 2
Multiplying/dividing by 2 shifts digits

19. Decimal -> Binary with Division
Step 1: Start with a blank answer and the number your are converting
Step 2: Divide your number by 2 to make a quotient and a remainder
Step 3: Place your remainder on the left side of your answer
Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2
Convert 105 to binary:
 Number ÷ 2 Q R
Answer
105

20. Decimal -> Binary with Division
Step 1: Start with a blank answer and the number your are converting
Step 2: Divide your number by 2 to make a quotient and a remainder
Step 3: Place your remainder on the left side of your answer
Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2
Convert 105 to binary:
 Number ÷ 2 Q R
Answer
105 ÷ 2 52 1

21. Decimal -> Binary with Division
Step 1: Start with a blank answer and the number your are converting
Step 2: Divide your number by 2 to make a quotient and a remainder
Step 3: Place your remainder on the left side of your answer
Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2
Convert 105 to binary:
 Number ÷ 2 Q R
Answer
105 ÷ 2 52 1

22. Decimal -> Binary with Division
Step 1: Start with a blank answer and the number your are converting
Step 2: Divide your number by 2 to make a quotient and a remainder
Step 3: Place your remainder on the left side of your answer
Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2
Convert 105 to binary:
 Number ÷ 2 Q R
Answer
105 ÷ 2 52 1
52 ÷ 2

23. Decimal -> Binary with Division
Step 1: Start with a blank answer and the number your are converting
Step 2: Divide your number by 2 to make a quotient and a remainder
Step 3: Place your remainder on the left side of your answer
Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2
Convert 105 to binary:
 Number ÷ 2 Q R
Answer
105 ÷ 2 52 1
52 ÷ 2 26 01

24. Decimal -> Binary with Division
Step 1: Start with a blank answer and the number your are converting
Step 2: Divide your number by 2 to make a quotient and a remainder
Step 3: Place your remainder on the left side of your answer
Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2
Convert 105 to binary:
 Number ÷ 2 Q R
Answer
105 ÷ 2 52 1
52 ÷ 2 26 01
26 ÷ 2 13
001

25. Decimal -> Binary with Division
Step 1: Start with a blank answer and the number your are converting
Step 2: Divide your number by 2 to make a quotient and a remainder
Step 3: Place your remainder on the left side of your answer
Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2
Convert 105 to binary:
 Number ÷ 2 Q R
Answer
105 ÷ 2 52 1
52 ÷ 2 26 01
26 ÷ 2 13
001
13 ÷ 2 6
1001
6 ÷ 2 3
01001
3 ÷ 2
101001
1 ÷ 2

26. Decimal -> Binary with Division
Step 1: Start with a blank answer and the number your are converting
Step 2: Divide your number by 2 to make a quotient and a remainder
Step 3: Place your remainder on the left side of your answer
Step 4: If your quotient is 0, you are done Otherwise, make the quotient your new number and go back to step 2
Convert 105 to binary:
 Number ÷ 2 Q R
Answer
105 ÷ 2 52 1
52 ÷ 2 26 01
26 ÷ 2 13
001
13 ÷ 2 6
1001
6 ÷ 2 3
01001
3 ÷ 2
101001
1 ÷ 2
128 64 32 16 8 4 2 1
= 105

27. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101

28. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
0 * 2

29. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101
0 * 2 + 1

30. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1

31. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1
1 * 2

32. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1 01
1 * 2 + 1

33. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1 01 3

34. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1 01 3
3 * 2

35. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1 01 3
3 * 2 + 0

36. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1 01 3 6

37. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1 01 3 6
6 * 2

38. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1 01 3 6 _
6 * 2 + 1

39. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1 01 3 6 _ 13

40. Binary -> Decimal with Multiplication
Step 1: Start with the number your are converting and the answer of 0
Step 2: Multiply your answer by 2
Step 3: Remove the leftmost digit of number and add it to your answer
Step 4: If number has no more digits, you are done Otherwise, go back to step 2
Convert to decimal:
 Number
Answer
1101
101 1 01 3 6 _ 13
128 64 32 16 8 4 2 1
= 13