1. 1.5 Adding Real Numbers Pre-AlgebraGreg Hood

2. What are real numbers?

3. Rational Numbers: Any number that can be written in the form of. As a decimal they repeat or terminate. ex: Repeats ex: Terminates Irrational Numbers: ex:  and These must be represented by a symbol (ex:  ), or as a rounded number, or in radical form because the decimal doesn’t repeat or terminate (stop). Integers: Whole numbers and their opposites (this means positive and negative whole numbers). ex: … ־ 4, 3, 2, 1, 0, 1 ־, 2 ־, 3 ־, 4 … Whole Numbers: Natural Numbers and zero. ex: 0,1,2,3… Natural or Counting Numbers ex: 1,2,3,4,… REAL NUMBERS

4. So what isn ’ t a real number? When you divide by zero and get no solution ( ), and = i (imaginary numbers). A rational number is any number you can write as a quotient of two integers, where b is not zero.

5. Write problem. What is the distance from zero? 7 Evaluate the expression. The absolute value of a number is its distance from zero on a number line. The symbol | a | represents the absolute value of a. Note: Distance cannot be negative. Example 3 Evaluate the expression. 1. Write problem. 2. What is the distance from zero?16

6. –|3| – 3 Example 4 Evaluate the expression. Example 5 Evaluate the expression. –|–29| – 29    

7. Use mental math to solve the equation. |x| = 25 Think. What numbers are 25 units from zero? 25, -25

8. Example 6 Use mental math to solve the equation. |x| = 1.7 Think. What numbers are 1.7 units from zero? 1.7, -1.7

9. Example 7 Use mental math to solve the equation. |x| = -4.2 Think. no solution Absolute value is never negative!

10. Down Down is negative. Speed tells you only how fast an object is moving. It can only be positive so it is the absolute value of velocity. UP UP is positive. Velocity tells you how fast an object is moving and in what direction. Example 8 A parachutist descends at a rate of about 17 feet per second. a. What is the parachutist ’ s velocity?–17 b. What is the parachutist ’ s speed?17 Speed is never negative. Motion is downward.

11. Write the value of these pieces. This piece has a value of 1. I will add more pieces. Write a numerical expression and simplify. Example 1 Let yellow = positive

12. Write the value of these pieces. This piece has a value of –1. I will add more pieces. Write a numerical expression and simplify. Example 2 Let red = negative

13. Step 1 ADD Step 2 ATTACH the common sign –8 + (–5) same To add two numbers with the same sign: Rules of Addition = – 13 8 + 5= 13 If the common sign is positive, the + is not written!

14. Write a numerical expression and simplify. Example 3 Let yellow = positive Let red = negative This expression has a special name. Do you know what it is? zero pair Adding or subtracting zero pairs will not affect the value of an expression.

15. When you add two numbers with opposite signs, Let yellow = positive Let red = negative you will need to subtract (take away) the zero pairs.

16. Write the value of these pieces. I will add more pieces. Write a numerical expression. Example 4 Let yellow = positive Let red = negative To simplify remove all the zero pairs. The pieces that remain model the simplified answer.

17. Write the value of these pieces. I will add more pieces. Write a numerical expression. Example 5 Let yellow = positive Let red = negative To simplify remove all the zero pairs. The pieces that remain model the simplified answer.

18. Step 1 SUBTRACT Step 2 ATTACH the sign of the number with the larger absolute value. 4 + (–9) opposite To add two numbers with the opposite signs: Rules of Addition –5–5

19. Example 6 Write a numerical expression and simplify. Can you tell at a glance what the sign is for the answer ? or

20. Example 7 Write a numerical expression and simplify. Can you tell at a glance what the sign is for the answer ? or

21. 1. Write problem. add 2. Signs alike – add write the common sign. –42 Example 8 Find the sum. Example 9 Find the sum. 1. Write problem. subtract 2. Signs unlike – subtract write sign of the largest absolute value. –8

22. Commutative Property: The order in which numbers are added does not change the sum. Associative Property: The way numbers are grouped when added does not change the sum. (a + b) + c = a + (b + c) Properties of Addition (2 + 5) + 10 = 2 + (5 + 10) x + y = y + x 9 + 4 = 4 + 9 Closure Property: The sum of any two real numbers is a unique real number. 5 + 3 = 8

23. Identity Property: The sum of a number and 0 is the number. Inverse Property: The sum of a number and its opposite is 0. x + 0 = x–16 + 0 = –16 When two signs are needed use parentheses or a superscript! Properties of Addition

24. 1. Write problem.12 + – 36 = – 36 + 12 2. Name the property.commutative Example 10 Name the property shown by the statement. Example 11 Name the property shown by the statement. 1. Write problem.0 + – 8 = –8 2. Name the property.identity

25. Example 12 Name the property shown by the statement. 1. Write problem.(4 + 5) + 6 = (5 + 4) + 6 2. Name the property.commutative Example 13 Name the property shown by the statement. 1. Write problem. – 7 + 7 = 0 2. Name the property.inverse

26. 1. Write problem.(9 + 6) + 5 = 9 + (6 + 5) 2. Name the property.associative Example 14 Name the property shown by the statement.

27. Example 15 Find the sum. 1. Write problem. 2. Follow the rules for the order of operations to simplify. If you override the left to right rule, you must show the step which gives support to your thinking!

28. Example 16 Find the sum. 1. Write problem. 2. Follow the rules for the order of operations to simplify. If you override the left to right rule, you must show the step which gives support to your thinking!

29. Example 17 Find the sum. 1. Write problem. 2. Use associative property to regroup. Then simplify.