Transcript ppt

MATH 1000 /11
Chapter 1
1.2 Symbols and Set of Numbers
1.3 Fractions
Sets
A set is collection of objects, each of
which is called a member or element of
the set.
Notation: A pair of brace symbols { }
encloses the list of elements is a the
set of elements containing that set of
elements.
MATH100/05/ Dr. H. Melikyan
Sets of numbers
•
•
•
•
Natural numbers – {1, 2, 3, 4, 5, 6 . . .}
Whole numbers – {0, 1, 2, 3, 4 . . .}
Integers – {. . . –3, -2, -1, 0, 1, 2, 3 . . .}
Rational numbers – the set of all numbers that can
be expressed as a quotient of integers, with
denominator  0
• Irrational numbers – the set of all numbers that can
NOT be expressed as a quotient of integers
• Real numbers – the set of all rational and irrational
numbers combined
•
A Number Line used to represent ordered real numbers has
negative numbers to the left of 0 and positive numbers to the
right of 0.
• Order Property for Real Numbers indicates how to use inequality
signs.
If a and b are real numbers,
a < b means a is to the left of b on a number line.
a > b means a is to the right of b on a number line.
• Absolute value of a number is the distance of that number away
from 0. a 0, since distances are non-negative.
| -7 | = 7, |- 0.13| = 0.13, | 3| = 3.
Section 1.3
Fraction is a quotient of two numbers.
• Numerator is the top number.
• Denominator is the bottom number.
Simplifying fractions (lowest terms)
• Involves factoring numerator and denominator into
prime numbers (natural numbers other than 1 whose
only factors are 1 and itself : 2, 3, 5, 7, 13, 37 …).
• A natural number bigger 1 , that is not prime is called
Composite number ( 15, 21, 1222)
MATH100/05/ Dr. H. Melikyan
Fundamental Principle of Fractions
• Can cancel common factors in numerator and
denominator.
• If a, b, c are real numbers such that b and c  0.
a c a

bc b
MATH100/05/ Dr. H. Melikyan
Example
Simplify the following fractions.
30
2 35
5
5



48 2  2  2  2  3 2  2  2 8
22
2 11

45 3  3  5
Since there are no common terms, the
fraction is already simplified.
12
2 23
1


60 2  2  3  5 5
MATH100/05/ Dr. H. Melikyan
Multiplying fractions
• Multiply numerators and denominators.
a c ac
 
b d bd
b, d  0
Dividing fractions
• Invert the divisor fraction (the reciprocal of divisor ).
• Then multiply the fractions.
a c a d ad
   
b d b c bc
MATH100/05/ Dr. H. Melikyan
b, d, c  0
Adding and subtracting fractions
• Required to have the same denominator.
• Have to change fractions to equivalent ones
until they have same denominator.
• Then combine the numerators, denominator will
be the common denominator.
a c ac
 
b b
b
a c ac
 
b b
b
b0
b0
MATH100/05/ Dr. H. Melikyan
Example
Add the following fractions.
3
9
223
12


 3

20 20 20
225
5
Subtract the following fractions.
7 5
7  6 5  5 42 25 17
 




10 12 10  6 12  5 60 60 60
MATH100/05/ Dr. H. Melikyan
MATH100/05/ Dr. H. Melikyan