week 7 modular numbers

8/14/2019 Week 7 Modular Numbers

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Discrete Mathematics is part of mathematics thatdeals with sets of objects that can be counted orprocesses that consist of a sequence of individual steps


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Natural numbers The sets of objects is called finite if the cardinality of

the set is a natural number.

This famous image has been digitized by computerand used as cover for Finite Mathematics textbook.


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True or False?

4 + 2 = 6

11 + 2 = 1


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Clock ArithmeticWith mathematical system based on a 12-hour clockExample :

Appointment at 4:00 pm but you late two hours andarrive at 6:00 pm, so if it is 11:00 am and arrive at1:00pm. That is , on a 12-hour clock;

4+2 = 6 and 11 + 2 = 1


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Addition on a 12-hour clock+ 1 2 3 4 5 6 7 8 9 10 11 12

1

2

34

5

6

78

9

10

11


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Addition on a 12-hour clock+ 1 2 3 4 5 6 7 8 9 10 11 12

1 2 3 4 5 6 7 8 9 10 11 12 1

2 3 4 5 6 7 8 9 10 11 12 1 2

3 4 5 6 7 8 9 10 11 12 1 2 34 5 6 7 8 9 10 11 12 1 2 3 4

5 6 7 8 9 10 11 12 1 2 3 4 5

6 7 8 9 10 11 12 1 2 3 4 5 6

7 8 9 10 11 12 1 2 3 4 5 6 78 9 10 11 12 1 2 3 4 5 6 7 8

9 10 11 12 1 2 3 4 5 6 7 8 9

10 11 12 1 2 3 4 5 6 7 8 9 10

11 12 1 2 3 4 5 6 7 8 9 10 11

Table 1


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Exercise 1:

Find clock sums 7 + 5

9 + 5

4 + 11 12 + 3


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Exercise 1:

Find clock sums 7 + 5 = 12

9 + 5 = 2

4 + 11 = 3 12 + 3 = 3


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Elementary operations Subtraction : a b = x means a = b + x

Multiplication :a x b = ab means b+b+b++b

Zero multiplication :if a = 0, then a x b =0 x b=0 Division : ab = a = x means a = bx provided b has

ban inverse for multiplication.

Plus a times


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Table of modulo 4+ 0 1 2 3

0 0 1 2 3

1 1 2 3 02 2 3 0 1

3 3 0 1 2

x 0 1 2 3

0 0 0 0 0

1 0 1 2 32 0 2 0 2

3 0 3 2 1


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Exercise 2Carry out the given operations on a 12-hour clock byusing the definitions for the elementary operations

a. 4 9 b. 4 x 9 c. 4 7 d. 4 .9


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Answer

a) 4 9 = x means 4 = 9 + x. What number when addedto 9 produces 4 ? (from Table 1, x = 7)

b) 4 x 9 means 9+9+9+9 = 12 (From Table 1)

c) 4 7 = t means 4 = 7t. The number that multipliedby 7 to obtain the result 4? (By trial and error:7x1=7,7x2=2,7x3=9,7x4=4)Therefore. T = 4.

d) 4 . Means 49 = s or 4 = 9s. Proof this !!9


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Modulo Five Arithmetic

+ 0 1 2 3 4

0

1

23

4

0 1 2 3 4

0

1

23

4

Consider a mathematical systembased on a 5-hour clock, numbered0, 1, 2, 3 and 4.

Table 2


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Instead of speaking about arithmetic on the 5-hourclock mathematicians usually speak of modulo 5arithmetic. The set {0,1,2,3,4}, together with theoperations in table 2 is called modulo 5 or mod 5system.

Eg. Suppose it is 4 oclock on a 5-hour clock. Whattime will the clock show 9 hours later?


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Eg. Suppose it is 4 oclock on a 5-hour clock. Whattime will the clock show 9 hours later?

Answer:We could write

4 + 9 = 3 and 2 + 1 = 3

Thus 4 + 9 = 2 + 1. Since we do not wish to confuse this

with ordinary arithmetic, then we use the followingnotation:

4 + 9 2 + 1, (mod 5)

Which is read 4 + 9 is congruent to 2 + 1, mod 5


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Exercise 3:

Checking Congruence

Decide whether each statement is true or false.

a) 3 8 (mod 5) b) 3 53 (mod 5)

c) 3 19 (mod 5)


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Answer1. Decide whether each statement is true or false.

a) 3 8 (mod 5) because 8 3 = 5, and 5 multiple of 5

b) 3 53 (mod 5), 53 3 =50 and 50 is multiple of 5c) 3 19 (mod 5), 19 3 = 16 and 16 is not a multiple

of 5


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Another way to determine whether two numbers arecongruent mod m is to divide each by mand checkthe remainders. If the remainders are the same, thenthe numbers are congruent mod m.

Eg. 3 5 gives a remainder 3,and 53 5 gives a remainder of 3,

so 3 53, 5 .


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Exercise 4:

Solve modular equationsSolve each equation for x.

a) 4 + 9 , 5 b) 15 + 92 , ( 5)

c) 2 + 4 , ( 5)

d) 2 4 , ( 5)

e) 7 x 5 , ( 7)

f) 3 - 5 , ( 12)


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Answera) 4 + 9 = 13 3, 5

b) 15 + 92 = 107 2, ( 5)

c) 2 + 4 = 6 1, ( 5)d) 2 4 = 7 4, ( 5)

3, ( 5)

e) 7 x 5 = 35 0, ( 7)

f) 3 - 5 = 15 5, 12 10, ( 12)


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Modular ArithmeticBased on 12-hour clock :

9 + 7 = 16 and 16 12 = 1 with remainder 4



15 2 = 13 and 13 12 = 1 with remainder 1

Exercise 5

True or false ?

a) 16 4 (mod 12) b) 12 0 (mod 12)

c) 23 8 (mod 12) d) 29 4 (mod 12)


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Answer 19 = 7, (mod 12)

37 = 1, (mod 12)

47 = 11,(mod 12 52 = 4, (mod 12)

108 = 0, (mod 12)

230 = 2, (mod 12)


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Exercise 7 (Level 2)1. Assume that today is Tuesday (1 Mac 2012). Draw themodulo 7 table for the month of Mac. Then, determinethe day of

a) 9 Mac

b) 22 Mac

c) 31 Mac


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AnswerDay Tue Wed Thur Fri Sat Sun Mon1 2 3 4 5 6 0

a) 9 Mac , 9 2, 7 9 7 = 2Thus, 9 Mac is Wednesday

b) 22 Mac, 22 1 7 22 7 = 1Thus, 22 Mac is Tuesday


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2. Assume that today is Monday (day 2). Determine theday of the week it will be at the end of each of thefollowing periods. (Assume no leap years.)

a) 24 days

b) 155 days

c) 365 days

d) 2 years


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Mind opener

Find some resources from books and journal about

Modular Arithmetic and History of Numbers Make a copy and keep inside your folio

Chinese calendar

Rat : 0,12,24,..Ox : 1,13,25.

week 7 modular numbers

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