Where do we go from here? What to do with all those numbers?

How many numbers do we have? We have 20 rows by 20 columns. Each cell is a number between 0 to 255. We have a row between 1 to 20. A column between 1 to 20. And a cell with a number between 0 to 255.

How many numbers do we have? We have 400 numbers between 0 to 255. What does it mean? What is a number anyway?

How do you learn number? We do not learn Seven (7) in the beginning. How do you describe a 7

We do this in the beginning One, two, three, four, … with fingers

What is counting? We do, 1, 2, 3, 4, … How do you describe this act? Counting

How to count? We use fingers, toes and digits. But we have to stop at 20. What can we do afterwards?

Remember these

A Quiz Can we count more than 10 with 2 hands?

A Quiz Yes, we can count more than 10 with 2 hands.

Numbering System The Hindu-Arabic Numerals 1, 2, 3, 4, 5, 6, 7, 8, 9 Concept of zero comes later. We have to tell the difference between 51 and 501.

Natural Number 1, 2, 3, 4, 5, … The positive integers. It is so natural.

Natural Number If you have two baskets, one contains apples and the other oranges, what does it mean when we say they have the same number of fruits. Try to do this at home. Whenever you take one apple out from the first basket, take an orange out from the other. When the baskets empty at the same time, they have the same number of fruits. We can say, there is a one-one correspondence between the basket of apples and the basket of oranges.

What is counting? Working on an one-one correspondence between a basket of fruits and the Natural Number. By the time we empty the basket, the count (number) of fruits in the basket in that Natural Number we arrive at. What if sometimes we cannot stop?

When will we stop? The Natural Number 1, 2, 3, 4, … will not stop. For every number you say, we can find another one comes after it. What do we mean ‘come after it’?

Come after what? We used to say greater than. It is a relationship between two Natural Numbers. It defines the Order of number. Given two numbers, a and b. Either a comes after b or b comes after a, otherwise a and b are equal.

The Order If we have a sequence 3, 5, 7, 12, 10, 8, … We can say the FIRST one is 3; The SECOND one is 5; The THIRD one is 7; Etc.

Counting Again Consider the list A = 1, 2, 3, 4, 5, 6, … And the list B = 2, 4, 6, 8, 10, 12, … And the list C = 1, 3, 5, 7, 9, 11, … We can always find an one-one correspondence among list A, B and C. That means all the 3 lists have the same count of numbers. What if we add the list B and C together? It gives the list A. What is the count now?

Some Operations Intuitively, we can do +, -, *, / upon the Natural Number without difficulty. ‘-’ calls upon the concept of Negative Number. ‘/’ requires a different kind of number. 2 / 3 is not a Natural Number. It is a Fraction.

Rational Number p / q is a Rational Number. If p and q are mutually prime, p / q cannot reduce to a Natural Number. 1/2, 2/3, 55/79 are Rational Number. The question is Can we count all the Rational Number with a form like p / q ?

Anymore Number?

Irrational Number Can the square root of 2 be a Rational Number?

Irrational Number Another common number.

Real Number How many of them? How dense are they? Can we count them one by one?

Real Number Consider the real numbers between 0 and 1. How many? How dense?

Real Number Try this out.

What the Fuck? Why should I know about this?

Analog vs. Digital You are told that our world is analog; the computer is digital. What does it mean? Traditionally, we model our world using analog means which is similar to a real number line between 0 and 1. In order to visualize it, however, we need to convert it to a digital way for display.

Being Digital Now go back to the self portrait photo. Remember the photo is 20 x 20 blocks. We can count from 1 to 20, which is the Natural Number. Between pixel 1 and 2, there is nothing in between. Although the photo is 2 dimensional, it can be converted to a 1 dimensional list of numbers. Remember the timetable exercise in class 1.

Being Digital Each block is a number between 0 to 255. Each number, say 167, denotes the brightness. We can say, 200 is brighter than 100, which uses the come after relationship of numbers. If two adjacent numbers differ greatly, we can notice a visible edge.

Sampling / Digitizing Your face is a smooth tone of sophisticated colours, i.e. the real numbers. It is represented by 20 x 20 numbers of brightness information, i.e. the natural numbers. This process is sampling / digitization. A mathematical process to produce a sequence of numbers, through +, -, *, /, % and others. It is where creativity comes into picture.

Information Visualization Let’s go back to Phil. If you are given a number 7, how can you present it?

Information Visualization 7

Seven

Information Visualization 笨

Position in 2D plane Size (width, length) Value Colour (HSB model) Pattern

Half-toning

Visualizing Text

Visualizing Lyrics

What else? Other than sampling, what else can we do? In illustration and animation, we often do not sample but draw the material. Can we draw from scratch with numbers?

Drawing with Numbers Yes, but how? An example,

Drawing with Numbers The 3 x 3 magic square with grey values

Drawing with Numbers The 3 x 3 magic square with HSB colour model.

Drawing with Numbers The 3 x 3 magic square with pattern.

Drawing with Numbers Try a Latin Square this time

Drawing with Numbers Latin Square with HSB colour model.

Filling a Square Fill up a square with linear number sequence

Filling a Square Fill up a square with linear number sequence

Filling a Square Fill up a square with linear number sequence

Filling a Square Fill up a square with linear number sequence

Filling a Square Fill up a square with linear number sequence

Any more Creativity? You do not have to use the Natural Number sequence. 1, 3, 5, 7, 9, 11, … 1, 2, 3, 5, 7, 11, … 1, 4, 9, 16, 25, 36, … 1, 3, 6, 10, 15, 21, … 1, 2, 6, 24, 120, 720, … 1, 2, 3, 5, 8, 13, …

Going to Infinity? What happen when the number grows too big? Remember the modulo operator learnt in primary school. For example 27 % 10 = 7

Simple Exercise Construct a number sequence through your own creation. Make at least 25 numbers. Restrict the number values within the range of 0 to 9. Fill up a square of 5 x 5 by the number using any creative method. Use grey scale, HSB colour model or a set of 10 patterns to fill up the square.