1. The Binary Number System
Data Representation

2. What is a number?

3. What is a number?
A number is a unit of an abstract mathematical system subject to the “Laws of Arithmetic.”

4. The Laws of Arithmetic
Succession
Addition
Multiplication

5. Number Categories Natural (Whole) Negative Rational
The counting numbers
Negative
Less than 0
Rational
An integer, or the quotient of 2 integers

6. Succession

7.  Positional Notation
The Decimal system is based on the number of digits we have. 
Positional Notation allows us to count past 10 by organizing numeric digits in columns.
Each column of a number represents a power of the base.
The base is 10.
The exponent is the order of magnitude for the column.

8. Positional Notation
103
102
101
100
10001
1001 11
The exponent is the order of magnitude for the column.
The Least Significant digit is in the right-most column.
The Most Significant digit is in the left-most column.

9. Positional Notation
103
102
101
100
10001
1001 11
The base is 10.

10. Positional Notation
103
102
101
100
10001
1001 11
The magnitude of the column is base exponent

11. Positional Notation
=27916
Consider a number like the one above.
How many does it represent?

12. Positional Notation
=27916
The size of a number is determined by multiplying the magnitude of the column by the digit in the column and summing the products.

13. Positional Notation
=27916
The columns are labelled with their exponents.

14. Positional Notation
=27916
The base of the system is 10.

15. Positional Notation
=27916
The magnitude of the column is base exponent

16. Positional Notation
*2 *7 *9 *1 *6
=27916
Multiply the magnitude of the column by the digit in the column.

17. Positional Notation
*2 *7 *9 *1 *6
27 thousand, 9 hundred, sixteen
Sum the products.

18. Binary Numbers
The binary number system is a means of representing quantities using only 2 digits:
0 and 1.
Like other number systems it’s based on
Positional Notation.

19. Positional Notation
In Binary, the columns have the expected exponents, 23 22 21 20 81 41 11

20. Positional Notation
In Binary, the columns have the expected exponents,
but the base of the system is 2. 23 22 21 20 81 41 11

21. Positional Notation
In Binary, the columns have the expected exponents,
but the base of the system is 2.
So the column magnitudes are powers of 2. 23 22 21 20 81 41 11

22. Binary
Rather than referring to each of the numbers as a binary digit, we shorten the term to bit.
To facilitate addressing, binary values are typically stored in units of 8 bits, which is called a byte.
Large values occupy multiple bytes.

23. A Single Byte
=255

24. A Single Byte
=255

25. A Single Byte
=255

26. A Single Byte
=255
is the largest decimal value that can be expressed in 8 bits. How many different patterns are there?

27. A Single Byte =0
There is also a representation for zero, making 256 (28) combinations of 0 and 1, in 8 bits.

28. Natural Numbers in Binary
Consider the pattern:
To calculate the Decimal equivalent:
multiply each digit by the value of the column
sum the products.

29. Natural Numbers in Binary

30. Natural Numbers in Binary

31. Natural Numbers in Binary

32. Natural Numbers in Binary

33. Natural Numbers in Binary
=149

34. Natural Numbers in Binary
Conversion from Decimal to Binary uses the same technique, in reverse.
Consider the value 73.
In base 10, this is 7 units of 10, plus 3 units of 1.

35. Natural Numbers in Binary
We need to express the value in terms of powers of 2. 27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1

36. Natural Numbers in Binary
What is the largest power of 2 that is included in 73? 27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1

37. Natural Numbers in Binary
64 is the largest power of 2 that is included in 73, so a 1 is needed in that position 27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1

38. Natural Numbers in Binary
Subtracting 64 from 73 leaves 9, which cannot include 32, or 16, but does include 8. 27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1

39. Natural Numbers in Binary
Subtracting 8 from 9 leaves 1, which cannot include 4, or 2, but does include 1. 27 26 25 24 23 22 21 20
128 64 32 16 8 4 2 1

40. Natural Numbers in Binary
So the 8 bit binary representation of 73 is:

41. Short Forms

42. Longer Numbers
Since 255 is the largest number that can be represented in 8 bits, larger values simply require longer numbers.
For example, is represented by:

43. Longer Numbers
Since 255 is the largest number that can be represented in 8 bits, larger values simply require longer numbers.
For example, is represented by:
Can you remember the Binary representation?

44. Short Forms for Binary
Because large numbers require long strings of Binary digits, short forms have been developed to help deal with them.
An early system was called Octal.
It’s based on the 8 patterns in 3 bits.

45. Short Forms for Binary - Octal
111 7
110 6
101 5
100 4
011 3
010 2
001 1
000
can be short-formed by dividing the number into 3 bit chunks (starting from the least significant bit) and replacing each with a single Octal digit.

46. Short Forms for Binary - Octal
111 7
110 6
101 5
100 4
011 3
010 2
001 1
000
added

47. Short Forms for Binary - Hexadecimal
0111 7
1111 F
0110 6
1110 E
0101 5
1101 D
0100 4
1100 C
0011 3
1011 B
0010 2
1010 A
0001 1
1001 9
0000
1000 8
It was later determined that using base 16 and 4 bit patterns would be more efficient.
But since there are only 10 numeric digits, 6 letters were borrowed to complete the set of hexadecimal digits.

48. Short Forms for Binary - Hexadecimal
0111 7
1111 F
0110 6
1110 E
0101 5
1101 D
0100 4
1100 C
0011 3
1011 B
0010 2
1010 A
0001 1
1001 9
0000
1000 8
can be short-formed by dividing the number into 4-bit chunks (starting from the least significant bit) and replacing each with a single Hexadecimal digit.

49. Short Forms for Binary - Hexadecimal
0111 7
1111 F
0110 6
1110 E
0101 5
1101 D
0100 4
1100 C
0011 3
1011 B
0010 2
1010 A
0001 1
1001 9
0000
1000 8

50. Short Forms for Binary Octal and Hexadecimal are number systems.
It is possible to perform arithmetic in both.
There are 64 (82) rules of octal addition, and 256 (162) rules of hexadecimal addition.
But why design a machine with so many rules when conversion to Binary is simple and there are only 4 rules of Binary addition?