Download Operations with Integers and Rational Numbers Note

MPM2D – MATHEMATICS REVIEW – PART 1
A.
OPERATIONS (ADDITION, SUBTRACTION, MULTIPLICATION,
DIVIDION) WITH INTEGERS
Integers are all the numbers that are positive and negative.
I = { ... , –3, –2, –1, 0, +1, +2, +3, ... }
1. Addition:
- if the signs are the same, then the sum has the same sign as well
Ex.
(12)  (5)  17
(7)  (5)  12
- if the signs are different, then the sum takes the sign of the larger number
Ex.
(18)  (5)  13
(7)  (2)  5
2. Subtraction:
- add the opposite
Ex.
(15)  (8)  15  8
 7
3. Multiplication and Division:
- if the two integers have the same sign, the answer is + (positive)
Ex.
(6)  (8)  48
(6)  (5)  30
- if the two integers have different signs, the answer is – (negative)
Ex.
(12)  (2)  6
(18)  (2)  9
4. BEDMAS (Order of Operations)
B
E
D
M
A
S
brackets ()
exponents X2
division /
multiplication x
addition +
subtraction -
Ex.
8  (4)  4  2 2
 8  4  4  2 2
 8  4  4  4
 2  1
 1
B.
OPERATIONS WITH RATIONAL NUMBERS
Rational Numbers are numbers that can be written as fractions.
a

Q =  a , b,   , b  0 
b

a
“The set of all numbers
such that
b
a, b are integers, and b  0 ”
1. Addition and Subtraction:
- to add or subtract fractions, you need to find a common denominator
- add or subtract numerators, denominator stays the same
- then reduce to lowest terms
- if you have a mixed fraction, you need to first change it to an improper
fraction and then find a common denominator, add or subtract, and
reduce to lowest terms
3 1

4 3
Ex.1
3 1

5 2
Ex.2

9
4

12 12

6 5

10 10

13
12

1
10
2. Multiplication:
- multiply the numerators and then multiply the denominators
- reduce to lowest terms
- if it is a mixed fraction, you need to first change it to an improper
fraction
a c ac
 
b d bd
Ex.
2 4 8
 
3 5 15
3. Division:
- multiply by the reciprocal
- reduce to lowest terms
- if it is a mixed fraction, you need to first change it to an improper
fraction
a c a d
  
b d b c

Ex.
2 4 2 5
  
3 5 3 4
ad
bc

10
12
4. Changing a Mixed Fraction to an Improper Fraction:
- multiply the whole number by the denominator
- add the numerator
- put this over the denominator
Ex.1
2

3
5
13
5
Ex.2
7

3
7
 52
7