Lesson 2 Operations with Rational and Irrational Numbers NCSCOS Obj.: 1.01; 1.02 Objective TLW 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, …} -5 5 Integers = {…, -2, -1, 0, 1, 2, …} Whole Numbers = {0, 1, 2, …} Natural Numbers = {1, 2, 3, …}

SUBTRACT and use the sign of the larger number. Addition Rule 1) 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 = ? -4 -2 2 4 Answer Now

-6 + (-3) = ? -9 -3 3 9 Answer Now

The additive inverses (or opposites) of two numbers add to equal zero. Example: The additive inverse of 3 is -3 Proof: 3 + (-3) = 0 We will use the additive inverses for subtraction problems.

What’s the difference between 7 - 3 and 7 + (-3) ? 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) 11b + (+2b) Same Signs --> Add and keep the sign. 13b

Which is equivalent to -12 – (-3)? 12 + 3 -12 + 3 -12 - 3 12 - 3 Answer Now

7 – (-2) = ? -9 -5 5 9 Answer Now

Review 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.

Absolute Value of 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 = 10 -100 = 100 5 - 8 = -3= 3

|7| – |-2| = ? -9 -5 5 9 Answer Now

|-4 – (-3)| = ? -1 1 7 Purple Answer Now

Line up the decimals and add (same signs). -2.564 Find the sum. 1) -2.304 + (-0.26) Line up the decimals and add (same signs). -2.564 2) Get a common denominator and subtract.

Find the difference. 3) Change subtraction to adding the opposite. Get a common denominator. Subtract and keep sign of the larger number.

Find the difference. 4) Change subtraction to adding the opposite. Get a common denominator and subtract.

5) Solve 6.32 – y if y = -3.42 Substitute for y: 6.32 - (-3.42) 6.32 + 3.42 9.74

Find the solution 6.5 – 9.3 = ? -3.2 -2.8 2.8 3.2 Answer Now

Find the solution . Answer Now

A rational number is a number that can be written as a fraction. How can these be written as a fraction? 3 =

Inequality Symbols

Ordering Rational Numbers 2 ways to order from least to greatest Get a common denominator Change the fractions to decimals (numerator  demoninator)

Which rational number is bigger? 1) Get a common denominator. 2) or convert the fraction to a decimal. 0.363 < 0.375 4 1 < 3 8

Which rational number is bigger? 7 4 or 1 6 Get a common denominator or convert to a decimal 1.75 < 1.83 42 24 < 44

Which symbol makes this true? 5 7 __ 3 4 < > = Answer Now

Which symbol makes this true? -2 -1 __ 9 4 < > = Answer 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

Multiplying fractions: Examples 1) (3x)(-8y) -24xy 2) Write both numbers as a fraction. Cross-cancel if possible. Multiplying fractions: top # • top # Bottom # • bottom# = -16

When multiplying two negative numbers, the product is negative. True False Answer Now

When multiplying a negative number and a positive number, use the sign of the larger number. True False Answer Now

3) = =

Multiply: (-3)(4)(-2)(-3) 72 -72 36 -36 Answer Now

an easy way to determine the sign of the answer 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? 3 negative signs -> Odd -> answer is negative -480

Positive or negative answer? Positive - even # of negative signs (4) Last one! 5) Positive or negative answer? Positive - even # of negative signs (4) Write all numbers as fractions and multiply. =12

What is the sign of the product of (-3)(-4)(-5)(0)(-1)(-6)(-91)? Positive Negative Zero Huh? Answer 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. True False Answer Now

When dividing a negative number and a positive number, use the sign of the larger number. True False Answer Now

The reciprocal of a number is called its multiplicative inverse. The reciprocal of is where a and b  0. 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 The reciprocal of is Example #2 The reciprocal of -3 is

Basically, you are flipping the fraction! We will use the multiplicative inverses for dividing fractions.

Which statement is false about reciprocals? Reciprocals are also called additive inverses A number and its reciprocal have same signs If you flip a number, you get the reciprocal The product of a number and its reciprocal is 1 Answer Now

Examples 1) When dividing fractions, change division to multiplying by the reciprocal.

2)

What is the quotient of -21 ÷ -3? 18 -18 7 -7 Answer Now

. Answer Now