Goals

Distinguish between categories of numbers
Describe positional notation
Convert numbers in other bases to base ten
Convert base-ten numbers to other bases
Describe the relationship between base 2, 8 and 16
Explain the importance to computing of bases that are powers of two

Number Categories

Numbers
- A unit in an abstract mathematical system
- Must follow the rules of that system (+,-, next, previous, ...)
Natural numbers, 0 and any number that can be derived from adding 1 to a natural number
Positive numbers - Natural numbers - {0}
Negative numbers - Opposite of the positive numbers
Integers - natural numbers and the negative numbers
Rational numbers - any number that is expressed as a ratio of two numbers
Irrational numbers - any number that cannot be expressed as a natural number
- PI
- a link

Natural Numbers

4² = 4*4 = 16
5³ = 5*5*5 = 125
6⁰ = 1
x^k = x*x*x ... *x (k times)
What is 836?
- 800 + 30 + 6
- 8 * 10² + 3 * 10¹ + 6 * 10⁰
In this example, 10 is the base
The base of a number system is the foundational value of a number system.
- In base 10, we use 10 digits (0,1,2,3,4,5,6,7,8,9)
- Notice the connection between fingers, and the basic numbers in the system
- In other bases we use different symbols
- For example in base two we use the digits (0,1)
- And base 8 we use the digits (0,1,2,3,4,5,6,7)
- In base 16, we add the letters A,B,C,D,E,F to the 10 numerals
When we want to represent a number at the base or higher, we go to a positional number system
- This involves storing information by where the digits are places
- A system of expressing numbers in which the digits are arranged in succession, the position of each digit has a place value and the number is equal to the sum of the products of each digit by its place value
- As in the example above
  - 1536 = 1 * 10 ³ + 5 * 10²+3*10¹+6*10⁰
  - 1536 = 1000 + 500 + 30 + 6
- An example in another base (base 4)
  - 312₄ = 3 * 4 ² + 1 * 4¹ + 2 * 4⁰
- Binary
  - Base two
  - Why?
  - Digits 0,1
  - 110101₂ = 1*2⁵+1*2⁴+0*2³+1*2²+0*2¹+1*2⁰ = 1*32+1*16+0*8+1*4+0*2+1*1 = 53
  - To convert a number in binary to base ten, just do this.
  - To convert a number in base ten to binary
```
While the quotient is not zero
   divide the decimal number by the new base
   add the remainder as the next digit of the new number (on the left)
   replace the number with the quotient
	   
```
  - ```
      207  <- quotient
    ------
  3 | 624
      6
      -
      02
      00
      --
       24
       21
       --
        3 <- remainder
  	   
```
- Convert 49 to base two
```
  49/2 = 24 R 1
  24/2 = 12 R 0
  12/2 =  6 R 0
   6/2 =  3 R 0
   3/2 =  1 R 1
   1/2 =  0 R 1
      Read this backwords, so the number is 110001
         
```
- check this 110001 = 1 * 2⁵+1*2⁴+1*2⁰ = 32+16+1
- Powers of two.
  - 2⁰ = 1
  - 2¹ = 2
  - 2² = 4
  - 2³ = 8
  - 2⁴ = 16
  - 2⁵ = 32
  - 2⁶ = 64
  - 2⁷ = 128
  - 2⁸ = 256
  - 2⁹ = 512
- Base 8 (Octal)
  - Binary numbers tend to get quite long (too many digits)
  - We often use octal to represent numbers
  - The nice part is, if a number is in binary, we can convert it to base 8 by grouping the digits in sets of three, adding 0 to the left end if necessairy
  - Binary to octal
    - 000₂ = 0 ₈
    - 001₂ = 1 ₈
    - 010₂ = 2 ₈
    - 011₂ = 3 ₈
    - 100₂ = 4 ₈
    - 101₂ = 5 ₈
    - 110₂ = 6 ₈
    - 111₂ = 7 ₈
  - 10110101001₂ = 010 110 101 001₂ = 2651₈
  - Octal to binary is easy as well
  - 25427₈ = 010 101 100 010 111₂
- Base 16 (Hexadecimal, Hex)
  - Same as octal, but groups of four
    - 0000₂=0₁₆, 1000₂=8₁₆
    - 0001₂=1₁₆, 1001₂=9₁₆
    - 0010₂=2₁₆, 1010₂=A₁₆
    - 0011₂=3₁₆, 1011₂=B₁₆
    - 0100₂=4₁₆, 1100₂=C₁₆
    - 0101₂=5₁₆, 1101₂=D₁₆
    - 0110₂=6₁₆, 1110₂=E₁₆
    - 0111₂=7₁₆, 1111₂=F₁₆
  - 3AF₁₆ = 0011 1010 1111₂
  - 10111001₂ = 1011 1001₂ = B9₁₆
- Adding in base 10
  - What is 9+3
  - What is 243 + 952
  - What is the carry ?
  - How is subtraction different?
- Adding in base 2
  - What is 10111+ 11010
  - Addition table in base 2
    
    + 0 1
    
    0 00 01
    
    1 01 10
Ethical Issues
- Digital Divide
  - Have/Don't have computers
  - Have/Don't have internet access
  - Have/Don't have training
  - Quality of each of these is an issue
  - Both in this country and around the world
- Does this make a difference?
  - In terms of educational opportunity?
  - In terms of education (success, access, ...)
  - In terms of employability
  - In terms of life/lifestyle?
- How invasive/ubiquitous are computers?
- What will happen as the years go by?
- Is this a problem?
  - If so, how can it be addressed?
  - If not, will it ever become one?