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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 Information Visualization Seven Information Visualization 笨 Information Visualization Information Visualization Information Visualization Information Visualization Information Visualization Information Visualization Information Visualization Information Visualization • • • • • Position in 2D plane Size (width, length) Value Colour (HSB model) Pattern Half-toning Visualizing Text Visualizing Text Visualizing Lyrics Visualizing Lyrics 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, 4 9 2 3 5 7 8 1 6 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. 1 2 3 4 3 4 1 2 4 3 2 1 2 1 4 3 Drawing with Numbers • Latin Square with HSB colour model. Filling a Square • Fill up a square with linear number sequence. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Filling a Square • Fill up a square with linear number sequence. 1 2 3 4 12 13 14 5 11 16 15 6 10 7 9 8 Filling a Square • Fill up a square with linear number sequence. 1 2 3 4 8 7 6 5 9 10 11 12 16 15 14 13 Filling a Square • Fill up a square with linear number sequence. 1 2 6 7 3 5 8 13 4 9 12 14 10 11 15 16 Filling a Square • Fill up a square with linear number sequence. 1 15 14 4 12 6 7 9 8 10 11 5 13 3 16 2 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.