The Open Office Impress slides that I've used in the final project of GTR.

1. GTR Final Project [9A(13)]
Mathematics Mini-lecture
Modulus Arithmetic and Field

2. But … What's 'Modulus Arithmetic'?
We all know about division...
a = bq+r (0≤r<b)
...Where q stands for 'quotient' and r for 'remainder'
Now, Lets define a mod b as follows...
a mod b = r ↔ a = bq+r (0≤r<b)
… Which is the action of finding remainder
Then we could define congruence...
a≡b (mod m) ↔ a mod m = b mod m
…Which is equal to say the remainders are the same

3. Exercise 1 (10s)
1.1) Calculate the following
1.1.a) 10 mod 10
1.1.b) 9 mod 8
1.1.c) 100 mod 65
1.2) Find x
1.2.a) 100 ≡ x (mod 10), 55<x<65
1.2.b) 95 ≡ x (mod 87), 180<x<190
1.2.c) 54≡ x (mod 8), 15<x<25

4. Exercise 1 (Answer)
1.1a) 0
1.1b) 1
1.1c) 35
1.2.a) 60
1.2.b) 182
1.2.c) 22

5. What about 'field'?
To know what is field, we must have to know:
set, group, abelian group and ring
'Set' is a group of numbers:
e.g. {1,2,3,4,5,...} = N
{1,3,5,7,9,...} = set of odd numbers
{1,4,9,16,...} = set of square numbers
To confirm that you understand, it is...
Exercise
...time now!

6. Exercise 2 (10s)
2.1) Define the following sets.
2.1.a) {2,4,6,8,10,...}
2.1.b) {2,4,8,16,32,...}
2.1.c) {1,2,3,4,5,6,7,8,9,10}
2.2) Write down the the following sets.
2.2.a) The set of odd numbers between 10 and 20
2.2.b) The set of triangular numbers
2.2.c) The set of multiples of 5 less than 101

7. Exercise 2 (Answer)
2.1a) Even numbers
2.1b) Positive powers of 2
2.1c) Natural numbers less than 11
2.2.a) {11,13,15,17,19}
2.2.b) {1,3,6,10,15,21,28...}
2.2.c) {5,10,15,20,...,90,95,100}

8. Then... What's 'group'?
We then have to define 'operation' ■ as...
a ■ b is an element of set G
...where a and b are elements of G.
And we all know the associative law...
(a ■ b) ■ c = a ■ (b ■ c)
...which means ''order doesn't affect the result''
Next is 'identity element' e ...
a ■ e = e ■ a = a
...which's like 0 for addition and 1 for multiplication

9. Then... What's 'group'?
At last is 'inverse element'...
a ■ b = b ■ a = e
...here we say b is the inverse element of a
Finally to define 'group' as...
1. Related to operation ■ and enclosed
2. Associative law is true for all elements
3. Identity element exists
4. Inverse element exists for all elements
...a set that satisfy all these postulates is a group
Note: in (1), a set is related to operation ■ and
enclosed means that a ■ b is an element of set G

10. Exercise 3 & Answer (10s)
Check if the following are groups...
a.) N, operation + (addition)
b.) N, operation * (multiplication)
c.) set of odd numbers, +
d.) set of even numbers, +
Answers:
a.) yes
b.) no (no inverse element)
c.) no (not closed, odd + odd = even)
d.) yes

11. After that... What's 'abelian group'?
We all know the commutative law...
a ■ b = b ■ a
...which is the only difference between a normal
group and an abelian group
An abelian group is...
a.) a group
b.) commutative law is true for all elements
...a set that satisfies both of the above postulates is
an abelian group

12. Exercise 4 & Answer (10s)
Check if the following are abelian groups...
a.) N, operation + (addition)
b.) set of 2*2 matrix, operation * (multiplication)
c.) set of odd numbers, +
d.) set of even numbers, +
Answers:
a.) yes
b.) no (not satisfy the commutative law)
c.) no (not closed, odd + odd = even)
d.) yes

13. Next... What's 'ring'?
In a group, we have only one operation...
...in a ring, we have TWO operations (■ and □)
We then have the associative law...
(a ■ b) □ c = (a □ c) ■ (b □ c)
Now, a 'ring' is...
a.) an abelian group with operation ■
b.) an abelian group with operation □ but NOT
NECESSARY to have inverse element
c.) associative law is true for all elements
...a set that satisfies all the above postulates is a ring

14. Exercise 5 & Answer (10s)
Check if the following are rings...
a.) N, operation + & *
b.) Q, operation * & +
c.) set of odd numbers, + & *
d.) set of even numbers, + & *
Answers:
a.) yes
b.) no (no inverse element for 1st
operation)
c.) no (not closed, odd + odd = even)
d.) yes (even * even = 4n)

15. Finally... What's 'field'?
In a ring, we maybe unable to do the reverse of □...
...in a field, we could do the reverse of □
A field is...
a.) a ring with operations ■ and □
b.) have the reverse element for operation □
(EXCEPT 0 when □ is multiplication)
...a set that satisfies both of the above postulates is a field
And it's EXERCISE TIME again!

16. Exercise 6 & Answer (10s)
Check if the following are fields...
a.) Q, operation + & *
b.) Z, operation + & *
c.) set of odd numbers, + & *
d.) R, operation + & *
Answers:
a.) yes
b.) no (no inverse element for 2nd
operation)
c.) no (not closed, odd + odd = even)
d.) yes

17. So... How are they related?
In a congruence, +, - & * does not change the congruence
If a≡b (mod m),
a + C ≡ b + C (mod m),
a - C ≡ b - C (mod m),
a * C ≡ b * C (mod m),
But not / …
15≡75 (mod 12),
(15/5)≡(75/5) (mod 12)
(15/3)≢(75/3) (mod 12)
Only if C's coprime with m, the congruence would hold

18. So... How are they related?
Now define {0,1,2,3,...,m-1} as a group with
operation ,⊠ which = (a / b) mod m, and denote the⊠
group 'Z/mZ'
We found that only when b is coprime with m, the
'group' actually holds.
Now define b m as 'b is coprime with m'⊥
Define (Z/mZ)
x
as the group of {x:0≤x<m,x m}with⊥
operation . This is the reduced residue class group.⊠

19. So... How are they related?
Now we could easily prove that (Z/mZ)
x
is an abelian
group. We could then define a ring with and ,⊞ ⊠
where is (a+b) mod m. The ring is called⊞
'residue class ring of modulo m'.
The following is the operation tables of the above ring:
⊞ 0 ... m-1
0 0 ... m-1
... ... ... ...
m-1 m-1 ... m-2
⊠ 0 ... m-1
0 0 0 0
... 0 ... ...
m-1 0 ... 1

20. So... How are they related?
When m is coprime to all elements in Z/mZ, then m is
prime. In this case, the residual ring of modulo m would
become a field. This field is called 'finite field of modulo p',
denoted as Fp, where Fp = Z/pZ.
Finite field is the final product of modulus arithmetic and
group theory. This thing is very useful in Maths, particularly
in the proof of Fermat's Last theorem, as you may see soon.
But before that, let's do some exercise first...

21. Exercise 7 & Answer (20s)
Write down the two operation tables of F3
Answers:
⊞ 0 1 2 ⊠ 0 1 2
0 0 1 2 0 0 0 0
1 1 2 0 1 0 1 2
2 2 0 1 2 0 2 1

22. How is finite field used?
Finite field is used for reduction of the 'elliptic curve'.
Substituting the points of the curve with the coordinates of
the finite fields, the resulting points that still holds is the
reduced curve.
Finite fields are used because the properties of it. The
rational field, Q, has an infinite numbers of elements, but
there's just 1 Q. However, there are infinite numbers of Fp,
but each has a finite number of elements. Therefore it is
used by mathematicians to explore the Q.

23. Why to reduce elliptic curves?
Finite field is used for reduction of the elliptic curves,
as mentioned above. But why?
Actually, this action is an important part in the proof of
the Fermat's Last Theorem by Andrew Wiles. However,
the proof is too difficult for talking here and is beyond the
scope of this lecture. What you need to know only is that,
there is a wide range of uses of finite field in number
theory.
So now, let's say...

24. Good
Bye!