Download Note01-Integer

Integer
SYSTEM IN NUMBER
OBJECTIVES
1. Students should be able to identify different numbers in numbering system.
2. Students should be able to use numbers for few appropriate operations.
3. Students should be able to calculate GCD and LCM correctly.
What, Which, Where, When
1. Knowledge about numbering system
Integer
(Clear / Not Clear )
Natural
(Clear / Not Clear ))
Real
(Clear / Not Clear )
2. Differentiate between numbers
Even integers
(Clear / Not Clear )
Odd integers
(Clear / Not Clear )
3. The usefulness of numbers
Dividing
(Clear / Not Clear )
Multiplying
(Clear / Not Clear )
4. Count
GCD
(Clear / Not Clear )
LCM
(Clear / Not Clear )
Kolman, Busby and Ross
page 20 – 31
Rosen 4th Ed
page 112 – 149
Rosen 5th Ed
page 53 - 195
Jonsonbaugh
page
Mattson
page 216 – 251
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Integer
NUMBER
-
Natural number, ℕ
-
Integer number, ℤ
-
Rational number, ℚ
-
Real number, ℝ
-
Complex number, ℂ
Integer, ℤ
One of the numbers in numbering system. Integer is widely used in calculation.
Fact 1
If n is a positive integer, then there are n integer i, where 1  i  n.
Fact 2
If m and n are positive integers with m  n, then there are n – m + 1 integer i, where m
in
Even integer is when it is twice (two times) any other integers.
Ex 1 :
-8, -6, -4, -2, 0, 2, 4, 6, 8
Odd integers are other than even.
Ex 2 :
-7, -5, -3, -1, 1, 3, 5, 7
Multiple and Divisor
If n and m are integers and n > 0, we can have m = qn + r, q and r are integers, 0 ≤ r ≤ n.
Ex 3 :
a) If n is 3 and m is 16, then 16 = 5(3) + 1, where q is 5 and r is 1
b) If n is 10 and m is 3, then 3 = 0(10) + 3, where q is 0 and r is 3.
c) If n is 3 and m is 9, then 9 = 3(3) + 0, where q is 3 and r is 0.
If r = 0 (as in Ex3), it is said that m is a multiple of n, written as n|m, read as n divides m. If
n|m, then m = qn and n ≤ |m| or m ≥ n for q times.
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Integer
In other words,
a) m is divisible by n;
b) n is a divisor for m;
c) n is a factor for m;
Ex 4 :
6 = 2(3)
where
m = qn
6 is a multiple for 3
6 can be divided by 3
3 divide 6
3 is a divisor for 6
3 is a factor for 6
so 3|6 = 2 or 6/3 = 2
Consider Ex 3(a).
Where n =3, and m = 16, and in this case, r ≠ 0.
For cases in which r ≠ 0, the integer part for q cannot be found. However, we have to
identify the integer number; either it is lower integer or upper integer.
We write the lower integer as m/n and the upper integer as m/n.
Ex 5 :
16/3 = 5
16/3 = 6
Consider Ex 3 (b). Find its lower integer and upper integer respectively.
Theorem 1
For all integer n > 1, n is a prime number if and only if its divisors are 1 and n.
(1 is not a prime number)
Theorem 2
All positive integer n > 1, can be written uniquely as p1k1 p2k2 p3k3 … psks, where p1 < p2 < p3
< … < ps are distinct primes that divide n. k’s are positive integers giving the number of
times each prime occurs as a factor of n.
Ex 6 :
6 = 2. 3
9 = 3. 3 = 32
24 = 8. 3 = 2. 2. 2. 3 = 23. 3
3
30 = 2. 15 = 2. 3. 5
Integer
Greatest Common Divisor
Let say a, b, and k are positive integers, and k|a and k|b, we say that k is a common
divisor for a and b. There might be a few k, in which the largest is d. d is called the
greatest common divisor (GCD) of a and b, written as
d = GCD (a, b).
Ex 7 :
Divisor for 18 are
1, 2, 3, 6, 9, 18
Divisor for 25 are
1, 5, 25
So, GCD (18, 25) = 1
Divisor for 24 are
1, 2, 3, 4, 6, 8, 12, 24
So, GCD (18, 24) = 6
Ex 8 :
168 = 2. 2. 2. 3. 7 = 23. 3. 7
192 = 2. 2. 2. 2. 2. 2. 3 = 26. 3
175 = 5. 5. 7 = 52. 7
So, GCD (168, 192) = 23. 31. 70 = 23. 3 = 24.
Find GCD (168, 175) and GCD (192, 175).
Least(Lowest) Common Multiple
Let say a, b, and k are positive integers, and a|k and b|k, we say that k is a common
multiple for a and b. There must be a few k (the one in common is ab), in which the
smallest is c. c is called the least common multiple (LCM) of a and b, written as
c = LCM (a, b).
Ex 9 :
LCM (168, 192) = 26. 31. 71
Find LCM (168, 175) and GCD (192, 175).
Exercise:
1. Find GCD (486, 70) and LCM(486, 70)
2. Find all information that you can get about Fibonacci; in person and its numbering
system.
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