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!
-1 + 3 = ?1. -4
2. -2
3. 2
4. 4
Answer Now
-6 + (-3) = ?1. -9
2. -3
3. 3
4. 9
Answer Now
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)
Which is equivalent to-12 – (-3)?
Answer Now
1. 12 + 3
2. -12 + 3
3. -12 - 3
4. 12 - 3
7 – (-2) = ?
Answer Now
1. -9
2. -5
3. 5
4. 9
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.
Examples
7 = 7
10
100
10 =
-100 =
5 - 8 = -3= 3
|7| – |-2| = ?1. -9
2. -5
3. 5
4. 9
Answer Now
|-4 – (-3)| = ?
Answer Now
1. -1
2. 1
3. 7
4. Purple
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.
Substitute for y: 6.32 - (-3.42)6.32 + 3.42
9.74
5) Solve 6.32 – y if y = -3.42
Find the solution6.5 – 9.3 = ?
Answer Now
1. -3.2
2. -2.8
3. 2.8
4. 3.2
Find the solution
Answer Now
2 1?
3 4
1
7
11
12
5
12
3
7
1. .
2.
3.
4.
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
Which symbol makes this true? 5
7__
3
41. <
2. >
3. =
Answer NowAnswer Now
Which symbol makes this true? -2
9__
-1
41. <
2. >
3. =
Answer NowAnswer Now
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
2
5
5
8
3)
10
40=
1
4=
Multiply: (-3)(4)(-2)(-3)
1. 72
2. -72
3. 36
4. -36
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 two negative numbers, the quotient is positive.
1. True
2. False
Answer NowAnswer Now
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
2) 1
5
3
10
1
5
10
3
10
15
2
3
What is the quotient of-21 ÷ -3?
1. 18
2. -18
3. 7
4. -7
Answer NowAnswer Now
1 10
?5 3
Answer NowAnswer Now
2
3
2
33
503
50
1. .
2. .
3. .
4. .