Slide 1 / 178 Slide 2 / 178 Algebra I The Number System & Mathematical Operations 2015-11-02 www.njctl.org Slide 3 / 178 Slide 4 / 178 Table of Contents Review of Natural Numbers, Whole Numbers, Integers and Rational Numbers Review of Exponents, Squares, and Square Roots Review of Natural Numbers, Review of Irrational Numbers Real Numbers Properties of Exponents Whole Numbers, Integers & Future Topics in Algebra II Rational Numbers Like Terms Click on a topic to Evaluating Expressions go to that section. Ordering Terms Glossary & Standards Return to Table of Contents Slide 5 / 178 Slide 6 / 178 Natural Numbers Natural Numbers Natural numbers were used before there was history. The first numbers developed were the Natural Numbers , also called the Counting Numbers. All people use them. 1, 2, 3, 4, 5, ... This "counting stick" was made more than 35,000 years ago and was found in Lebombo, Swaziland. The three dots, (...), means that these numbers continue forever: there is no largest counting number.. The cuts in this bone record the number "29." Think of counting objects as you put them in a container those are the counting numbers. http://www.taneter.org/math.html
Slide 7 / 178 Slide 8 / 178 Natural Numbers and Addition Numbers versus Numerals They were, and are, used to count objects Numbers exist even without a numeral, such as the number > goats, indicated by the cuts on the Lebombo Bone. > bales, > bottles, A numeral is the name we give a number in our culture. > etc. Drop a stone in a jar, or cut a line in a stick, every time a goat walks past. That jar or stick is a record of the number. Slide 9 / 178 Slide 10 / 178 Numbers versus Numerals Whole Numbers Adding zero to the Counting Numbers gives us the Whole Numbers . 0, 1, 2, 3, 4, ... If asked how many tires my car has, I could hand Counting numbers were someone the above marbles. developed more than 35,000 years ago. That number is represented by: It took 34,000 more years to 4 in our Base 10 numeral system · invent zero. IV in the Roman numeral system · 100 in the Base 2 numeral system · This the oldest known use of zero (the dot), about 1500 years ago. It was found in Cambodia and the dot is for the zero in the year 605. http://www.smithsonianmag.com/history/origin- number-zero-180953392/?no-ist Slide 11 / 178 Slide 12 / 178 Why zero took so long to Invent Why zero took so long Horses versus houses. Would I tell someone I have a herd of zero goats? 3 Or a garage with zero cars? 2 Or that my zero cars have zero tires? Zero just isn't a natural number, but it is a whole number. 1 0 zero horses = zero houses
Slide 13 / 178 Slide 14 / 178 Integers Whole Numbers Each time a marble is dropped in a jar we are doing addition. The below number line shows only the integers . A number line allowed us to think of addition in a new way. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -1 1 2 3 4 -10 -9 -8 -7 -6 -5 -4 -3 -2 0 5 6 7 8 9 10 Slide 15 / 178 Slide 16 / 178 Review: Fractions Review: Fractions There are an infinite number of fractions between each pair of integers, since the space between any two integers can be divided Remember that division led to a new set of numbers: fractions. by as large a number as you can imagine. Fractions are the result when you ask questions like: On the next slide, a few are shown, but in reality there are as many fractions between any pair of integers as there are integers. 1÷2 = ? 1÷3 = ? 2÷3 = ? 1÷1,000,000 = ? Fractions can be written as the ratio of two numbers: 1÷2 = ? asks the question: -2 / 3 , -1 / 4 , -1 / 8 , 1 / 3 , 4 / 5 , 7 / 5 , 80 / 4, etc. If I divide 1 into 2 equal pieces, what will be the size of each? Or in decimal form by dividing the numerator by the denominator: The answer to this question cannot be found in the integers. -0.666, -0.25, -0.125, 0.333, 0.8,1.4, 20, etc. New numbers were needed: fractions. The bar over "666" and "333" means that pattern repeats forever. Slide 17 / 178 Slide 18 / 178 Fractions Rational Numbers There are an infinite number of fractions between each pair of Rational Numbers are numbers that can be expressed as a integers. ratio of two integers. Using a magnifying glass to look closely between 0 and 1 on this This includes all the fractions, as well as all the integers, since number line, we can locate a few of these fractions. any integer can be written as a ratio of itself and 1. Fractions can be written in "fraction" form or decimal form. When written in decimal form, rational numbers are either: -1 1 2 3 4 -10 -9 -8 -7 -6 -5 -4 -3 -2 0 5 6 7 8 9 10 Terminating, such as 1 - = -0.500000000000 = -0.5 Notice that it's easier to find their location when they are in decimal 2 form since it's clear which integers they're between...and closest to. Repeating, such as 1 = 0.142857142857142857... = 0.142857 1 / 3 2 / 3 4 / 5 1 / 5 1 / 4 1 / 2 7 1 = 0.333333333333333333... = 0.33 3 0.20 0.33 1 0 0.66 0.80 0.25 0.50
Slide 19 / 178 Slide 20 / 178 Powers of Integers Just as multiplication is repeated addition, exponents are repeated multiplication. Review of Exponents, Squares, For example, 3 5 reads as "3 to the fifth power" = 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 and Square Roots In this case "3" is the base and "5" is the exponent. The base, 3, is multiplied by itself 5 times. Return to Table of Contents Slide 21 / 178 Slide 22 / 178 Special Term: Squares and Cubes Powers of Integers A number raised to the When evaluating exponents of negative numbers, keep in mind A number raised to the third power can be said to the meaning of the exponent and the rules of multiplication. second power can be said be "cubed." to be "squared." For example, (-3) 2 = (-3)(-3) = 9, That's because the volume That's because the area of is the same as (3) 2 = (3)(3) = 9. of a cube of length x is x 3 : a square of length x is x 2 : "x cubed." "x squared." However, (-3) 2 = (-3)(-3) = 9 x A = x 2 is NOT the same as -3 2 = -(3)(3) = -9, x V = x 3 x x Similarly, (3) 3 = (3)(3)(3) = 27 x is NOT the same as (-3) 3 = (-3)(-3)(-3) = -27, Slide 23 / 178 Slide 24 / 178 The Root as an Inverse Operation The Root as an Inverse Operation Inverses of exponents are a little more complicated for two reasons. Performing an operation and then the inverse of that operation returns us to where we started. First, there are two possible inverse operations. We already saw that if we add 5 to a number and then subtract 5, The equation 16 = 4 2 provides the answer 16 to the question: what we get back to the original number. is 4 raised to the power of 2? Or, if we multiply a number by 7 and then divide by 7, we get back One inverse operation is shown by: to the original number. This provides the answer 4 to the question: What number raised to the power of 2 yields 16? This shows that the square root of 16 is 4. It's true since (4)(4) = 16
Slide 25 / 178 Slide 26 / 178 Logs as an Inverse Operation 1 What is ? The other inverse operation will not be addressed until Algebra II. Just for completeness, that inverse operation is 2 = log 10 100. It provides the answer 2 to the question: To what power must 10 be raised to get 100. You'll learn more about that in Algebra II, but you should realize it's the other possible inverse operation. Slide 27 / 178 Slide 28 / 178 2 What is ? 3 What is the square of 15? Slide 29 / 178 Slide 30 / 178 4 What is ? 5 What is 13 2 ?
Slide 31 / 178 Slide 32 / 178 6 What is ? 7 What is the square of 18? Slide 33 / 178 Slide 34 / 178 Review of the Square Root of a Number 8 What is 11 squared? The following formative assessment questions are review from 8th grade. If further instruction is need, see the presentation at: http://njctl.org/courses/math/8th-grade/numbers-and- operations-8th-grade/ Slide 35 / 178 Slide 36 / 178 9 10 A 6 A 9 B -6 B -9 C is not real C is not real
Slide 37 / 178 Slide 38 / 178 (Problem from ) 11 12 A 20 B -20 C is not real Which student's method is not correct? A Ashley's Method B Brandon's Method On your paper, explain why the method you selected is not correct. Slide 39 / 178 Slide 40 / 178 13 14 Slide 41 / 178 Slide 42 / 178 15 16 A 3 A -3 B B C No real roots C D
Slide 43 / 178 Slide 44 / 178 Review of the Square Root of Fractions 17 The expression equal to is equivalent to a positive integer when b is The following formative assessment questions are review from 8th grade. If further instruction is need, A -10 see the presentation at: B 64 http://njctl.org/courses/math/8th-grade/numbers-and- C 16 operations-8th-grade/ D 4 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. Slide 45 / 178 Slide 46 / 178 18 19 C C A A no real solution no real solution B D B D Slide 47 / 178 Slide 48 / 178 21 C A B no real solution D
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