COUNTING

counting

The Thai Fruit Stall

How would you count all this fruit?
By the number of fruit for each type
By total weight, cost or sale value
How would you represent the quantity of all this fruit with a single meaningful number?
Only in units common to all: (Normally results in a real value and not an integer)
- weight
- total cost
- total sale value

Counting on fingers

counting

Identify major ways to count with fingers

one to one representation extended to both hands
finger order to display 10 units on one hand
one to one on working hand; group count on the other|

Do all cultures count with their hands the same way?
no
What is the maximum number that can represented by fingers?
Using one hand as the unit counter and the second hand as a completed group counter it is possible to represent 2 digits of data on 2 hands.
The square of the counting basis determines the maximum count possible: $5^2= 25; 10^2 = 100; 12^2= 144$

COUNTING SYSTEMS

Tallies

tally

How does this system work?
1 item is represented with 1 tick which are grouped in units of 5
How do you add and substract in this system?
add or erase tick mark, respectively
What is the key disadvantage of this representation?
run out of space with large number counts
hard to conceptualize and represent multiplication
How do you add and substract in this system? `
Add or erase a tick mark, respectively

Roman Numerals

Fill in this table:

Roman Numerial	Working	Decimal
MCMXLVII	1000 + 900 + 40 + 5 + 2	1947
DCCCLXXX	500 + 300 + 50 + 30	880
MMCMXCIX	2000 + 900 + 90 + 9	2999

What do all the letters mean?

Roman symbol	Decimal equivalent
I	1
V	5
X	10
L	50
C	100
D	500
M	1000
$\bar V$	5000
$\bar X$	10,000
$\bar L$	50,000
$\bar C$	100,000
$\bar D$	500,000
$\bar M$	1,000,000

What was the maximum number that could be represented?

\[\bar M\bar M\bar M\bar M\bar D\bar M\bar X\bar CM\bar XCMXCIX (4,999,999)\]

How do you add and substract in this system?
Carries and borrows require a change of symbol

\[\begin{array}{rcl} 39 + 27 & = & XXXIX + XXVII\cr &= & xxx\ v\ iiii + xx\ v\ ii\cr &= & xxxxx\ vv\ iiiiii\cr 66 &= & L X V I\cr & & \cr 52 - 29 & = & LII - XXIX \cr &=& xxxxx\ ii - xx\ v\ iiii\cr &=& xxxx\ vv\ ii - xx\ v\ iiii\cr &=& xxxx\ v\ iiiiiii - xx\ v\ iiii\cr 33 &=& XXIII\cr \end{array}\]

Is this system of counting still used?
Page numbering of the front matter in books
Numbering of clauses in legal documents
European and American

Decimal numbers

2 x 100 ------- 200
|   4 x 10 ----  40
|   |   3 x 1 -   3
|   |   |         
2   4   3       243

Why is this more efficient than tally?
Fewer symbols
Introduces a concept of digits that can be used to represent every power of 10
What is the role of the O numeral?
Represent zero or a null place holder
What do Arabic numerals actually look like?

Arabic numbers

Arithmetric

Addition and subtraction

How does addition and subtraction work?
- For single digit numbers addition will advance the count of the first number by the number given in the second number.
- Subtraction will reverse the count by the number given in the second number.
How does multi-column addition work?
We start with the units column and work column by column.
For addition, add the digits. If the sum $\ge$ 10, subtract 10 from the sum and carry the 1 to the next column.
For subtraction, if the result would be negective, subtract 1 from the next column and add 10 to the current column. Then the digits are subtracted and the result is recorded in the column.

      1  0  1 
         9  3
     ========
      1  9  4
     
            2  3
      5  7  4  1
    ============
      5  7  6  4
     
        3  5
        1  7
      ======
        4 12  Sum over loads the 1st column
      4+1  2  Shift the value of ten to the left column
        5  2  Final Answer
        
       7  8  3
       2  9  8
      =========
          1  1
       1  7
       9
      ========
    1  0  8  1

NUMBERS USING DIFFERENT BASES

Binary numbers - Base 2

Base: 2
Range of digits:0, 1
Values of columns: 1, 2, 4, 8, 16, 32, …

     1 x 32 -------------- 32
     |  0 x 16 ------------ 0
     |  |  1 x 8 ---------- 8
     |  |  |  1 x 4 ------- 4
     |  |  |  |  0 x 2 ---- 0
     |  |  |  |  |  1 x 1 - 1
     |  |  |  |  |  |        
     1  0  1  1  0  1      45

Binary addition

Determine the decimal equivalents.
$1010 + 111 = 10 + 7 = 17$
$10111 + 101 = 23 + 5 = 28$
Figure out the binary sums of these binary numbers

     1010  Rearrange in right justified format
     +111        
     ====
     1121  (not allowed, need to -2 and carry the one)
     1201  (not allowed, need to -2 and carry the one)
     2001  (not allowed, need to -2 and carry the one)
    10001  Final answer!!! (16+1 = 17)


    10111 
    + 101
    =====
    10212  (not allowed, need to -2 and carry the one)
    11020  (not allowed, need to -2 and carry the one)
    11100  Final answer!!! (16+8+4 = 28)

Octal numbers (Base 8)

Base: 8
Range of digits: 0, 1, 2, 3, 4, 5, 6, 7
Values of columns: 1, 8, 64, 512, 4096, …

     2 x 512 -------- 1024
     |  7 x 64 ------  448
     |  |  1 x 8 ----    8
     |  |  |  5 x 1 -    5
     |  |  |  |           
     2  7  1  5       1485

Hexadecimal numbers (Base 16)

Base: 16
Range of digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
Values of columns: 1, 16, 256, 4096, …

    10 x 4096 ------- 40960
    |  0 x 256 ------     0
    |  |  14 x 16 ---   224
    |  |  |  15 x 1 -    15
    |  |  |  |  
    A  0  E  F        41199

Describe how to do hexadecimal addition

          1    A    F  
        + 2    F    F
        ============== 
          3   25   30  <- Show column sums in decimal
          3 25+1   14  <- Readjust by -16 and carry 1
          3+1 10   14  <- Readjust by -16 and carry 1
          4   10   14  
          4   A    E  <- convert to Hexadecimal digits

       = 4AE

      3  5 
    + A  5
    ========
      13 10  (No adjustment needed; <16)
      D  A   (Replace by hexidecimal symbols  )
  
      = DA

There are 3 approaches commonly used:

Add the number as a standard column wise process starting with the rightmost column.
Convert to decimal and back convert to hexadecimal (usually slower)

\[\hbox{1AF}_{16} + \hbox{2FF}_{16} = 431_{10} + 767_{10} = 1198_{10}\] \[1198_{10} = \left(4 \times 256 + 10 \times 16 + 14 \times 1\right)_{10} = \hbox{4AE}_{16}\]

  3. Many calculators can to calculate in other bases so set the base to 16 and add the numbers.

\[\left(\hbox{1AF} + \hbox{2FF} = \hbox{4AE}\right)_{16}\]

Switching between bases

Complete this table

Decimal	Binary	Octal	Hexa- decimal
39	100101	45	25
42	101010	52	2A
25	11001	31	19
26	11010	32	1A

Describe the steps to convert numbering systems

For this example take the number 149 (decimal) and convert it to Octal
Determine the large power of 8 that is equal or less than the number. $8^2 = 64$
Divide the number by the power of 8. $149_{10} / 64_{10} = 2_{10}, Remain= 21_{10}$
Jot down the leading digit $2xx_8$ and divide the power value by 8 that is $64_{10} / 8 = 8_{10}$
Divide the Remainder by the new power value: $21_{10} / 8_{10}= 2_{10}, Remain=5_{10}.
Jot down the next digit $22x_8$ and divide the power value by 8 that is $8_{10} / 8_{10} = 1$
Since the power value is 1: put the remainder in the units place. $225_8$

Alternative number systems

1952 Myanmar Coinage

Old Kyat * 2pe = 0.05 kyat * 2pe = mu = 0.10 kyat * 2mu = mat = 0.20 kyat * 5 mat = 1 silver Kyat * 16 silver kyat = 1 gold kyat

New Kyat * 100 pe = 1 kyat * coins: 5,10,25,50 pyas

Convert the following:

Old Kyat to New Kyat:

\[\begin{array}{l} \hbox{1 gold kyat, 2 silver kyat, 4 mat, 1mu}\\ \qquad = 16 \times 1 + 2\times 1 + 4 \times 0.20 + 1 \times 0.10\\ \qquad = 16 + 2 + 0.80 + 0.10 = 18.90\\ \end{array} \]

New Kyat to Old Kyat:

\[\begin{array}{rclc} 27.40 &=& 27.40 -16 = 11.40 & 1 \hbox{ gold kyat}\\ & & 11.40 - 11 = 0.40 & 11 \hbox{ silver kyat} \\ & & 0.40 - 2\times 0.20 = 0 & 2 \hbox{ mat}\\ \end{array}\]

Egg shipments:

Thai packaging: * 10 * 30 * 300 * 1200

America produces 93,162,174,216 eggs each year. If one day’s production is shipped to Thailand what would the size of the shipment be in both standards?
American:

\[\begin{array}{rll} 93162174216 & = {93162174216 \over 1200} & = 77635145\ pallet\\ 216 &= 216/30 & = 7\ trays\\ 6 & & = 6\ eggs\\ \end{array}\]

Thai:

\[\begin{array}{rll} 93162174216 & = {93162174216 \over 1728} & = 53913295\ (12gross)\\ 456 &= {456 \over 144} & = 3\ gross\\ 24 & = {24 \over 2} & = 2\ dozen\\ \end{array}\]

Urdu counting:

Urdu Units	Decimal Equivalant	Scientific Notation
Hazar	1,000	$\huge 10^3$
Lakh	100,000	$\huge 10^5$
Crore	10,000,000	$\huge 10^7$
Arab	1,000,000,000	$\huge 10^9$
Kharab	100,000,000,000	$\huge 10^{11}$

Determine the size of the following populations using Urdu counting units:

Country	Arabic units	Urdu units
China:	1,388,336,022	1 Arab, 38 Crore, 83 Lakh, 36 Hazar, and 22
India:	1,342,512,706	1 Arab, 34 Crore, 25 Lakh, 12 Hazar, and 706
World:	7,524,885,124	7 Arab, 52 Crore, 48 Lakh, 85 Hazar, and 124

Factoring

Grouping 12 units

How many ways can 12 items be evenly grouped?
When was the concept of a dozen introduced?
Why do you think the concept of a dozen is so common?

Factoring of numbers

factors

\[\Large\begin{array}{cccccclcccccccl} \small\blacktriangle &\small\circ &[1] & = & 1 \times 1 & = & 1 & & & \small \bullet & [6] & = & 2 \times 3 \times 1 & = & 2 \times 3 \\ \small\blacktriangledown & \small\bullet & [2] & = & 2 \times 1 & = & 2 & & \small\blacktriangledown &\small\circ & [7] & = & 7 \times 1 & = & 7 \\ \small\blacktriangledown &\small\circ &[3] & = & 3 \times 1 & = & 3 & & & \small\bullet & [8] & = & 2 \times 2 \times 2 \times 1 & = & 2^3 \\ &\small\bullet &[4] & = & 2 \times 2 \times 1 & = & 2^2 & & & \small\circ & [9] & = & 3 \times 3 \times 1 & = & 3^2\\ \small\blacktriangledown &\small\circ &[5] & = & 5 \times 1 & = & 5 & & &\small\bullet & [10] & = & 2 \times 5 \times 1 & = & 2 \times 5 \\ \end{array}\]

\[\hbox{Key:}\ \ \ \ \blacktriangle\ \hbox{Identity}\ \ \ \ \blacktriangledown\ \hbox{Prime}\ \ \ \ \circ\ \hbox{Odd}\ \ \ \ \bullet\ \hbox{Even}\]

Factors

Determine the rule for labelling a number as odd or even.
Divisible by 2: even
Not divisible by 2: Odd
Determine a method for determining if a number is prime.
Identify all the prime numbers between 1 and 25.
What are the prime factors of these numbers?
- 27: 3 x 3 x 3
- 30: 2 x 3 x 5
- 45: 3 x 3 x 5
- 111: 3 x 3 x 3 x 3
- 210: 2 x 3 x 5 x 7

Real numbers

Define the following:
Rational numbers: Numbers that can be expressed as a fraction of integers
Irrational number: Numbers that cannot be expressed as a fraction of integers
Imaginary numbers: Numbers with a factor containing $\sqrt{-1}$

Converting real to integers

Floor(x) - red
Round(x) - blue
Ceiling(x) - green

NumberLines

Converting real to integers

Complete the table using these functions

Number	round(x)	ceiling(x)	floor(x)
-3.99	-4	-3	-4
-3.50	-4	-3	-4
-3.30	-3	-3	-4
-3.01	-3	-3	-4
1.00	1	1	1
1.45	1	2	1
2.99	3	3	2
3.01	3	4	3
4.50	5	5	4

Coordinate Systems

Cartesian Coordinate space

2D Representation

3D Representation

### 3D Space: with projects to 2D

coordinate

Polar Coordinates

polar

Geographic coordinates

PoundCake coordinate

Multiple coordinate space

Dimensions of a cake

8 ounce package cream cheese
1 1/2 cups butter
3 cups white sugar
6 eggs
3 cups all-purpose flour
1 teaspoon vanilla extract

PoundCake

What is the maximum number of dimensions that can be render in a graph on paper?
No limit, except for practical limits caused by the resolution of the drawing equipment.
Example:

Rose Diagram

How many dimensions are needed to describe variants of this pound cake recipe?
6
How would you graph these dimensions on paper?
Bar graph would work just fine.
How would you record these dimensions in a spreadsheet?
Each column represents the corresponding value for each dimension
Each row represents a specific instance or case.

IT100 Session 2: - Numbers

Answers key: Robert Batzinger

5 Sept 2017

COUNTING

The Thai Fruit Stall

Counting on fingers

COUNTING SYSTEMS

Tallies

Roman Numerals

Decimal numbers

Arithmetric

Addition and subtraction

NUMBERS USING DIFFERENT BASES

Binary numbers - Base 2

Binary addition

Octal numbers (Base 8)

Hexadecimal numbers (Base 16)

Switching between bases

Alternative number systems

1952 Myanmar Coinage

Convert the following:

Egg shipments:

Urdu counting:

Factoring

Grouping 12 units

Factoring of numbers

Factors

Real numbers

Converting real to integers

Converting real to integers

Coordinate Systems

Cartesian Coordinate space

2D Representation

3D Representation

Polar Coordinates

Geographic coordinates

Multiple coordinate space

Dimensions of a cake

Roman symbol	Decimal equivalent
I	1
V	5
X	10
L	50
C	100
D	500
M	1000
\(\bar V\)	5000
\(\bar X\)	10,000
\(\bar L\)	50,000
\(\bar C\)	100,000
\(\bar D\)	500,000
\(\bar M\)	1,000,000

Urdu Units	Decimal Equivalant	Scientific Notation
Hazar	1,000	\(\huge 10^3\)
Lakh	100,000	\(\huge 10^5\)
Crore	10,000,000	\(\huge 10^7\)
Arab	1,000,000,000	\(\huge 10^9\)
Kharab	100,000,000,000	\(\huge 10^{11}\)