1. WARM UP 1 Write fractional notation for each number 5.15 2 Write the decimal notation for 3 Calculate a. 6.79 + 3.4 b. 3 – 1.53 c. 2.21 x 1.8 d. 2.7 ÷ 7.2 e. f. g. h. 4. What is 25% of 25?130% of 64 5. Write the correct symbol =,. 0.074 _ 0.703 6. Write the correct ratio: a. 28 astronauts for 4 missions b. 52 wks in 12 mon 3.6 10.191.4 7 3.9780.375 6.25 83.2 < 7 to 1 13 to 3

2. ALGEBRA 1 REVIEW REAL NUMBERS & OPERATIONS

3. OBJECTIVES Identify rational numbers & distinguish between rational & irrational numbers Add positive & negative numbers Subtract positive & negative numbers

4. KEY TERMS & CONCEPTS  Absolute value  Additive inverse  Difference  Integers  Irrational numbers  Natural numbers  Number line  Opposite  Rational numbers  Real numbers  Subtraction  Subtrahend  Whole numbers  |x|  -x

5. REAL NUMBERS  The most important set of numbers in algebra is the set of real numbers.  There is exactly one real number for each point on a number line -2.5-½2 π  The positive numbers are shown to the right of zero and the negative to the left. Zero is neither positive nor negative.  The sets of natural numbers [1, 2, 3, 4,……..], whole numbers [0, 1, 2, 3, 4,……] and integers [-2, -1, 0, 1, 2, 3,….] are all subsets of the set of real numbers.

6. RATIONAL NUMBERS The real numbers consist of the rational and irrational numbers DEFINITION Rational numbers are those that can be expresses as a ratio, where a and b are integers and b ≠ 0 These are rational numbers: 4, 9.6, 0 Since they can be written as: 4/1, 96/10, 0/1

7. IRRATIONAL NUMBERS If a real number cannot be expressed as a ratio of integers 1/b, b ≠ 0, then it is called irrational. For instance, we can prove that there is no irrational number that is a square root of 2. We can come close but there is not rational number whose square is exactly 2. Thus, √2 is not a rational number. It is irrational. Unless a whole number is a perfect square, its square root is irrational.

8. IRRATIONAL NUMBERS The following numbers are irrational: √2, √8, -√45, √11, π. Decimal notation for a rational number either ends or repeats. Decimal notation for an irrational number never ends and never repeats.

9. EXAMPLES Determine which are rational and which are irrational numbers: 1. 8.974974974… 2. 3.12112111211112… 3. 4.325 4. √17 (numerals repeat) Since they repeat, the number is rational. We can express it as 8.974. (numerals do not repeat) Since they do not repeat and does not end the number is irrational. Since the number ends, it is rational. Since 17 is not a perfect square, it is irrational.

10. TRY THIS… a. 7.42 b. √49 c. 0.47646464… (numerals repeat) d. -√32 e. 59/37 f. 2.5734107656631…(numerals do not repeat)

11. ABSOLUTE VALUE The absolute value of a number a is the distance between a and 0 on a number line. The symbol represents the absolute value of a. If a is positive, then = a Example: = 2 If a is 0, then = 0= 0 If a is negative, then = -a = -(- 2) = 2

12. ABSOLUTE VALUE The absolute value of a number is its distance from 0 on a number line. We denote absolute value of x as |x|. - 6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 6 units4 units Since 4 is four units from 0, |4| = 4. Since -6 is six units from 0, |-6| = 6. The absolute value of 0, |0| = 0.

13. REAL NUMBERS ADDITION One way to add or subtract two real numbers is to use the number line. To add a positive number move right To add a negative number, move left - 6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

14. REAL NUMBERS ADDITION Recall the rules of signs for adding real numbers. RULES FOR ADDITION OF REAL NUMBERS 1. To add when there are like signs, add the absolute value. The sum has the same sign as the addends. 2. To add when there are unlike signs, subtract the absolute values. The sum has the sign of the addend with the greatest absolute value. Examples: Add -5 + (-9) = -14 23 + (-11) = 12 Adding absolute values; the sum is negative. Subtracting absolute values; the positive addend has greater absolute value

15. TRY THIS… a. -8 + (-9) b. -8.9 + (-9.7) c. -6/5 + (-23/10) d. 14 + (-28) e. -4.5 + (7.8) f. 3/8 + (-5/6)

16. ADDITIVE INVERSE Every real number has exactly one additive inverse or opposite. The additive inverse of a number is the number added to it to get 0. The additive inverse of x is –x. PROPERTY OF ADDITIVE INVERSES For every real number a, there is exactly one number b for which a + b = 0 The additive inverse of a number is the number opposite it, with respect to 0 on the number line.

17. ADDITIVE INVERSE  To find the additive inverse of 5, we reflect the opposite side of 0. the additive inverse of 5 is -5.  To find the additive inverse of a number quickly, just change the sign. The additive inverse of a number is the number opposite it, with respect to 0 on the number line.

18. REAL NUMBER SUBTRACTION Subtraction is defined in terms of addition. Subtraction and addition are inverse operations. SUBTRACTION The difference a – b is the number c such that c + b = a. We can always subtract by adding an inverse. The number subtracted is called the subtrahend. To subtract, we can change the sign of the subtrahend and then add it to the other number. 14 – 8 = 14 + (-8)

19. EXAMPLES 1) -12 – 19 = You can then use the number line to start at negative 12 and go left 19 units because you are adding a negative number. 2) 5 – (-4) = 5 + 4 = 9

20. TRY THIS… a. 8 – (-9) b. 23.7 – 5.9 c. -11/16 – (23/12)

21. CH. 1.1 HOMEWORK Textbook pg. 7 & 8 #12, 16, 18, 24, 28, 30 & 32