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###### tags: `Number Theory` `NT01`
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L01 Natural Number and Modular Arithmetics
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## This week
- Introduction: Module, Home coding, Friday
- Natural Number and (Modular) arithmetic
- Intro to Python (List, string, Dictionary)
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### 1. Natual Number
<font size = 4>Our early-kinds SMARTLY invent it for counting purpose, which was believed that, happend "before history"</font>.
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#### 1.1 Other set of Numbers
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#### 1.2 Initial Arithmetic: Addtion
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One rule attached to the Natural number was addtion. We might started by +1 on a certain number, but human realised that,
>create something can be momorised, will reduce the step of operatioins
so certain number system, the process of addtion can be much more simplified.
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-- E.g. (1+1+1+1+1+1) + (1+1+1+1+1) Compare with 6 + 5
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<font size=5>Features/properties of Addition in Natural numbers:</font>
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- Commutativity: a+b=b+a, for any a, b that are Natural numbers
- Associativity: a+(b+c)=(a+b)+c, for any a,b,c that are Natural numbers
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#### 1.25 let's repeat our smart idea again
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Now we have new problem: 9+9+9+9+9+9+9+9+9
>create something can be momorised, will reduce the step of operatioins
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-- E.g. 9+9+9+9+9+9+9+9+9 Compare with $9\times9 = 81$
Previously, out ancients generalise sticks/marks to number system.
Then, they generalise addtion to multiplication
Like a computer, they increase the important infomation stored in RAM, for a shorter calculation running.
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#### 1.3 Then: something interesting when "/" was introduced
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Division lead to a new problem: the remainder problem, because not every time, we apply two numbers into a division arithmetic, the output would be a integer...
e.g. 7/3 or 8/3
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Or,
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If I am asking you to describe relationship between two numbers, 5 and 26, what are you going to say?
1. 26 = 5 + 21, You can say that 26 is 5 plus 21
2. $26= 5 \times 5+1$, You can say that 26 is 5 of 5 sumup, plus 1
The first one is trivial but nothing interesting;
The second one is interesting in manny field of Mathematics.
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#### 1.35 Euclidean Division Lemma
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This is first time in this Module we come across the Great Euclid, You will meet him again shortly.
**Euclidean Division Lemma (easier version):**
> Given two positive integers a and b, with $b \ne 0$ and $a > b$, there exist unique integers q and r such that, $a = bq + r$, where $0\le r < b$
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*The proof of this lemma will be provided for interested reader after the lecture, as well as in Week 10's Math Appendix*
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Thanks for the Euclid's Inspiration, Now we have another way to describe relationship between 5 and 26;
3. $26 = 5\ mod\ 7$
The real world of Mathematics if far more interesting than calculation!
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2. Modular Arithematic
<font size = 4>Actually, it appears frequently when we talking about time.</font>
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