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Lesson 2 Operations with Rational and Irrational Numbers
NCSCOS Obj.: 1.01; 1.02
ObjectiveTLW State the coordinate of a point on a number line
TLW Graph integers on a number line.
TLW Add and Subtract Rational numbers.
TLW Multiply and Divide Rational numbers
The Number Line
Integers = {…, -2, -1, 0, 1, 2, …}
Whole Numbers = {0, 1, 2, …}
Natural Numbers = {1, 2, 3, …}
-5 0 5
Addition Rule1) When the signs are the same,
ADD and keep the sign.(-2) + (-4) = -6
2) When the signs are different,
SUBTRACT and use the sign of the larger number.
(-2) + 4 = 2
2 + (-4) = -2
Karaoke Time!Addition Rule: Sung to the tune of
“Row, row, row, your boat”
Same signs add and keep,different signs subtract,
keep the sign of the higher number,then it will be exact!
The additive inverses (or opposites) of two numbers add to equal zero.
-3
Proof: 3 + (-3) = 0
We will use the additive inverses for subtraction problems.
Example: The additive inverse of 3 is
What’s the difference between7 - 3 and 7 + (-3) ?7 - 3 = 4 and 7 + (-3) = 4
The only difference is that 7 - 3 is a subtraction problem and 7 + (-3) is an addition problem.
“SUBTRACTING IS THE SAME AS ADDING THE OPPOSITE.”
(Keep-change-change)
When subtracting, change the subtraction to adding the opposite (keep-change-change)
and then follow your addition rule.
Example #1: - 4 - (-7)- 4 + (+7)
Diff. Signs --> Subtract and use larger sign.3
Example #2: - 3 - 7- 3 + (-7)
Same Signs --> Add and keep the sign.-10
11b + (+2b)
Same Signs --> Add and keep the sign.
13b
Okay, here’s one with a variable!Example #3: 11b - (-2b)
1) If the problem is addition, follow your addition rule.
2) If the problem is subtraction, change subtraction to adding the opposite (keep-change-change) and then follow the addition rule.
Review
Absolute Valueof a number is the distance from
zero.
Distance can NEVER be negative!
The symbol is |a|, where a is any number.
Find the sum.1) -2.304 + (-0.26)Line up the decimals and add (same signs).
-2.564
2)5
8
3
4
Get a common denominator and subtract.
5
8
6
8
1
8
Find the difference.
3)5 3
9 5
3
5
5
9
25
45
27
45
2
45
Change subtraction to adding the opposite.
Get a common denominator.
Subtract and keep sign of the larger number.
Find the difference.
4)1
2
6
7
61
2 7
7
14
12
14
5
14
Get a common denominator and subtract.
Change subtraction to adding the opposite.
A rational number is a number
that can be written as a fraction.How can these be written as a fraction?
3 = 3
1
23
4
11
4
Inequality Symbols
Symbol Meaning < is less than > is greater than ≠ is not equal to ≤ is less than or equal to ≥ is greater than or equal to = is equal to
Ordering Rational Numbers
2 ways to order from least to greatest
1. Get a common denominator
2. Change the fractions to decimals (numerator demoninator)
Which rational number is bigger?
1) Get a common denominator.32
88
33
88
4
11or
3
8
2) or convert the fraction to a decimal.0.363 < 0.375
4
11<
3
8
Which rational number is bigger?
1) Get a common denominator
2) or convert to a decimal1.75 < 1.83
7
4
11
6
7
4or
11
6
42
24< 44
24
Multiplying Rules
1) If the numbers have the same signs then the product is positive.
(-7) • (-4) = 28
2) If the numbers have different signs then the product is negative.
(-7) • 4 = -28
Examples
1) (3x)(-8y)
20 4
5
2) Write both numbers as a fraction.
Cross-cancel if possible.
20
1
4
5
80
5= -16
Multiplying fractions:
top # • top #
Bottom # • bottom#
-24xy
When multiplying two negative numbers, the product is negative.
1. True
2. False
Answer NowAnswer Now
When multiplying a negative number and a positive number, use the sign of
the larger number.1. True
2. False
Answer NowAnswer Now
an easy way to determine the sign of the answer
3 negative signs -> Odd -> answer is negative-480
When you have an odd number of negatives, the answer is negative.
When you have an even number of negatives, the answer is positive.
4) (-2)(-8)(3)(-10)Do you have an even or odd number of
negative signs?
Last one!
5) 1
2
6 5 2
5
2
Positive or negative answer?Positive - even # of negative signs (4)
Write all numbers as fractions and multiply.
=12
1
2
6
1
5
1
2
5
2
1
12010
What is the sign of the product of (-3)(-4)(-5)(0)(-1)(-6)(-91)?
1. Positive
2. Negative
3. Zero
4. Huh?
Answer NowAnswer Now
Dividing Rules
1) If the numbers have the same signs then the quotient is positive.
-32 ÷ (-8 )= 4
2) If the numbers have different signs then the quotient is negative.
81 ÷ (-9) = -9
When dividing a negative number and a positive number, use the sign of the
larger number.1. True
2. False
Answer NowAnswer Now
The reciprocal of is
where a and b 0.
a
b
b
a
The reciprocal of a number is called its multiplicative inverse.
A number multiplied by its reciprocal/multiplicative inverse is
ALWAYS equal to 1.
Example #1
7
2.
2
7The reciprocal of is
2
7
7
2 1
11
Example #2
The reciprocal of -3 is
1
3.
1
11
3
1
1
3
Basically, you are flipping the fraction!
We will use the multiplicative inverses
for dividing fractions.
Which statement is false about reciprocals?
1. Reciprocals are also called additive inverses
2. A number and its reciprocal have same signs
3. If you flip a number, you get the reciprocal
4. The product of a number and its reciprocal is 1
Answer NowAnswer Now
When dividing fractions, change division to multiplying by the reciprocal.
Examples
1) 3
4
5
8
3
4
8
5
6
5
24
20