Number, Part I of II

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1 Lesson 1 Number, Part I of II 1 massive whale shark is fed while surounded by dozens of other fishes at the Georgia Aquarium. The number 1 is an abstract idea that can describe 1 whale shark, 1 manta ray, etc David E. Shormann Review: Shormann Algebra 1, Lessons 1 and 2* *New to Shormann Math? Please refer to these review lessons from Algebra 1 (found on the e-learning campus) any time you find a concept that you don t remember covering in your previous math course. Rules and Definitions Rules Rules that we use in math are about any things and some things, without specification of definite particular things. Alfred North Whitehead Numbers are rules. The main numerals we use in mathematics to represent numbers include 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Numbers are also ideas, and numerals are the symbols we use to describe numbers. Definitions - inductive reasoning: The process of discovering rules. Scientific endeavors are inductive. Discovery of new mathematical relationships requires inductive reasoning. - deductive reasoning: The process of applying rules. Mathematics is deductive. There are mathematical rules yet to be discovered. - numeral: A symbol or symbols used to express the idea of number. - abstract: dealing with properties and ideas of things, just because they are things, regardless of feelings, emotions, and sensations we might connect with them. An idea that can describe an infinite number of concrete objects. For example, the number 3 is an abstract idea that can describe 3 bears, 3 trucks, 3 cars, etc DIVE, LLC

2 - concrete: In mathematics, a word used to describe real objects. Abstract ideas are always based on real things. - digit: Any of the Hindu-Arabic numerals 1 through 9, and 0. - counting (natural) numbers: Numbers used to count objects. Ex. 1, 2, 3,... - whole numbers: Counting numbers and the number 0. - integers: Whole numbers, and all negative numbers. - real numbers: Any number used to describe a positive or negative number. Includes all integers, and all decimal numbers and fractions trichotomy axiom: For any two real numbers a and b, exactly one of the following is true: a < b, a = b, a > b Welcome Welcome to Shormann Algebra 2! If you completed Shormann Algebra 1, I have some good news for you. Recall that mathematics is deductive in nature, which means that the focus is to learn the rules and then apply them to the study of God s creation (science). And because math is deductive, the first quarter of any Shormann Math course is designed to introduce you to the most important rules you will use for the rest of the course. And since Algebra 2 is just application of (mostly) the same rules from Algebra 1, you won t have a lot of new content to learn during the first quarter. Whether you did Shormann Algebra 1 or not, you probably don t remember everything from your Algebra 1 course, so be wise, humble, and hard-working, and spend time reviewing when you need to. 1A What is mathematics? Before we begin our study of number, let s look at some ways of defining mathematics. Mathematics is many things wrapped into one, so it is not easy to have just one definition for it. Here are a few that famous mathematicians and math teachers have come up with over the years: The science which investigates the means of measuring quantity(l. Euler, Elements of Algebra, 1765). The foundation of exact thought as applied to natural phenomena(a.n. Whitehead, An Introduction to Mathematics, 1911). Mathematics is the classification and study of all possible patterns(w. W. Sawyer, Prelude to Mathematics, 1955). A study of space and quantity (Kline, Mathematics and the Physical World, 1959). Whitehead said mathematics is abstract, meaning it deals with properties and ideas of things just because they are things, regardless of feelings, emotions, and sensations we might connect with them. It doesn t matter how I feel about whale sharks; if there is 1 in a photo, then there is 1, and no other value

3 works. If I wrote 9 whale sharks are being fed in the photo above, you would immediately think that either I made a typo or I am lying. Math helps us know that if we search for truth, we can often find it. Because math reveals truth to us, it naturally leads to the question, Is there a Source of truth? In other words, mathematics points us to God. Only the fool says in their heart there is no God (Psalm 14:1). If you have foolish thoughts like this, and have not yet accepted God s saving gift of faith (Ephesians 2:8-9), please consider doing that right now! Believe and repent, then be baptized in the name of God s son, Jesus Christ, who gave His own life for the remission of sins, and you will receive the gift of the Holy Spirit (Acts 2:38-39). When we trust Christ as our Savior, we can know more things, but not every thing, about our origins, about what we are supposed to be doing right now, and about what will happen in the future (Proverbs 1:7). We can see the unity and diversity found in the Holy Trinity, where Scripture proclaims God is Father, Son and Holy Spirit. We see what is general (God) in what is particular (Father, Son, Spirit), and what is unchangeable in what is changing. We should not be surprised that God s attributes, like his divine nature and eternal power, are revealed in mathematics. Romans 1:20 says His invisible attributes like these are clearly seen in creation, so we have no excuse for denying God s existence. The word Godhead is also used to describe God s divine nature, reflecting the Trinity, which in turn reflects God s perfect example of unity and diversity. In Christ we see the fullness of the Godhead bodily (Colossians 2:9), meaning Christ is truly divine, but also truly human. God s divine nature is also abstract, because He exists apart from our feelings about Him. If our feelings, emotions, and sensations were required for God to exist, then He would be nothing more than an idol made by human hands. But Scripture is clear that God does not need us to create Him. He is the one who gives life and breath to all things (Acts 17:29). God has made it obvious that unity and diversity exist. Being able to describe it though can lead to confusion, and you should not feel the least bit ignorant if you don t fully understand how one thing can also be many. In fact, even though it s true, nobody completely understands it. God has revealed His attributes to us, but not yet fully (I Corinthians 13:12). Understanding some things, but not everything, about a concept requires humility, patience, joy, perseverance and prayer. And whether you are a Christian or not, let me be clear that attitudes of pride, arrogance, impatience and general grouchiness will not make learning math easier. To help you get a better understanding of things that relate to the concept of unity and diversity, study this table.

4 Unity God Jesus Christ General Abstract Eternal Infinite Continuous One Diversity Trinity (Father, Son, Spirit) Fully Divine, Fully Human Particular Concrete Mortal Finite Discrete Many Mathematics is considered the language of science. It is a tool for studying God s creation. Since God has an abstract nature, it makes sense that He would design our brains with the ability to use an abstract tool to help us learn about how His creation operates. Humans, and no other organisms, were created in God s image (Genesis 1:26). He designed us to be creative, like Him! And, as the famous scientist, Johannes Kepler said, when we study His creation, we think God s thoughts after Him, giving us, and no other organisms, the ability to manage His creation (Genesis 1:28). Mathematical tools, because they are abstract, supply just what is needed for a biblical, but also scientific description of creation. These tools will help you be the best you can be at using and managing God s creation. Do you want to be a good manager of His creation or a bad one? You should want to be a good manager, which means you never again need to ask what do I need to know math for? Let s finish this introduction to mathematics with one more definition of mathematics, which we will use for the rest of the course: Mathematics is the language of science and a God-given tool for measuring and classifying pattern and shape. David Shormann 1B A Brief History of Number A number is an idea. John Saxon What is number? A number is an abstract idea that applies in an infinite number of situations. We use numerals as symbols representing the idea of numbers. If something is abstract, it can be applied the same way to describe many things, like the number 2, or 2+2, etc. Earlier we discussed 1 whale shark. Whale sharks are real, but the number 1 is not. You cannot go to a farm and pluck 1 s off a tree like they were apples! Numbers, like math itself, are abstract rules. one-ness is an idea than can be applied to a variety of situations; 1 shark, 1 bear, 1 joyful thought, etc. It seems strange that one thing, a number, can describe many things. It is strange, but it obviously works! The abstract nature

5 of our Creator (one God, three persons) reminds us we should not be surprised to find aspects of His creation that have abstract qualities, too. Hindu-Arabic digits: Hindu-Arabic numerals arrived in western Europe at least by the 12 th century. They include the digits 0,1,2,3,4,5,6,7,8,9. Roman numerals: Developed by Romans in the centuries before Christ (B.C.), they are still used occasionally to mark dates, page numbers, and hours on a clock face. The symbols I, X, C, M stand for 1, 10, 100, and The symbols V, L, D stand for 5, 50, and 500. Normally, Roman numerals are added. Example 1.1 Convert the following Roman numerals to the Hindu-Arabic numeral system. a) CLXII b) DXC solution: a) CLXII = = 162. Note how, if you were given 162, you could find the corresponding Roman number by breaking 162 into 10 s, 5 s, and 1 s, and matching these values with the appropriate Roman numerals. b) If a smaller Roman numeral precedes a larger one, then subtract. For example, IV = 5-1 = 4. Likewise, IX = 9, XC=90, IC=99. So, DXC = = 590. Base 10, 2, etc. systems: The base of a system relates to its place value. For example, the base 10 system (also called the decimal numeral system) works in multiples of 10. As you can see in the example below, moving from right to left, each place is 10 times more than the place to its right = = = = =1/ =1/100 Use of the base 10 system dates back to at least 3000 B.C., with Egyptians using a heiroglyphic grouping system. A quick glance at a normal human s hands suggests that counting by 10 s was probably standard practice in the pre-flood world, too. Positional base systems have a number of digits equal to the base value. For example, the base 10 system has 10 digits, 0 through 9. The base 2 system uses 2 digits, 0 and 1. The base 2 system is important in computer applications, where the entire system is run by switching electric current on (1) or off (0) at different times and locations. A positional base 2 example is formed simply by switching out the 10 s in the base 10 example with 2 s:. 2 3 =8 2 2 =4 2 1 =2 2 0 =1 2-1 =1/2 2-2 =1/4 Example 1.2 Convert the binary number 1101 to base 10. solution: 1(8) + 1(4) + 0(2) + 1(1) = = 13.

6 An example of the binary system in use on a computer is with the color quality for a computer monitor. For example, the 16 bit quality, allows for 2 16 possible colors to be displayed. You will learn more about bits in Lesson 25. Arithmetic symbols: Our most common modern arithmetic symbols include + for addition, - for subtraction, or or () for multiplication, and for division. Most of these symbols were first used in arithmetic and algebra in the 15th-17th centuries. + and - are also used to describe positive and negative numbers, like +3 and -3. It is important to know that math is considered the language of science. It is a tool used to study God s creation. And like every other language, it has its own symbols. Sometimes, the symbols mean just one word, like + means plus, for example. However, some symbols have a meaning that is a phrase or even a sentence. During Shormann Algebra 2, you will learn many new symbols. It may be confusing at first, but be patient and persevere! Just like if you were learning a new language, you will naturally start memorizing what the symbols mean through repeated use. You will also see that using the symbols is faster than writing out the words or phrases corresponding to each symbol. 1C Types of Numbers Refer to the Lesson 1 definitions for different types of numbers. Be prepared to answer questions like the following: Example 1.3 True or False. All real numbers are also integers. solution: False. Real numbers include decimal numbers, and integers don t. It is true however, that all integers are real numbers. Example 1.4 True or False. All whole numbers are also counting numbers. solution: False. Whole numbers start with 0, but counting numbers start with 1, so all whole numbers are not counting numbers. Real numbers include all positive and negative numbers, with and without decimal places. This includes all numbers we can write as ratios of integers (rational numbers), along with all numbers that cannot be written that way (irrational numbers). Irrational numbers are unique because they have no recognizable repeating pattern of decimals, no matter how many decimal places you try to show. For example, the irrational number π, pronounced pie, rounded to 2 decimal places, equals Rounded to 20 decimal places, it equals Now, no one has ever

7 proven with 100% certainty that pi is irrational, but as far as we know, it has an infinite and non-repeating decimal pattern! You may also be thinking, if there are real numbers, are there imaginary numbers too? Well, actually, there are! They are also called complex numbers, and we ll discuss them in Lesson 2. First, let s continue with real numbers and discuss ways to organize them. Number lines are a useful tool for understanding the differences between number types, as well as positive and negative numbers. A number line is just a horizontal line, with or without arrowheads at the ends, divided into equal sections, like you see on a ruler. The number line is usually drawn with zero at the center, positive values to the right and negative values to the left. One way to think of negative numbers is that they are the opposite of a positive number. For example, -2 is the opposite of 2. If you look at the number line, it is easy to see that they are both 2 tick marks from 0, you just move in opposite directions from 0 to reach each value. Example 1.5 Organize the following numbers by graphing them on a number line: 4.1,2,6 3 4, 2 solution: Place a dot on the number line to identify the location of each number. This procedure is called graphing. Some points are more difficult to locate exactly, like 4.1. Just do your best to estimate its location. Finally, when working with inequalities, number lines are useful tools for gaining a better understanding of the different number types. Example 1.6 Graph the following on a number line. x 2, if a) Domain = Integers b) Domain = Counting Numbers solution: a) The domain refers to the set of acceptable values you can use. Since the domain is integers, we will use dots to represent the discrete integer values. Note the arrow on the end is also shaded to represent a continuing pattern:

8 b) Since the domain is counting numbers, we have to start with 1: Example 1.7 Graph the following on a number line. x 4, if a) Domain = Reals b) Domain = Whole numbers solution: a) The / through the inequality symbol means not, so, according to the trichotomy axiom from your Definitions, if x is not greater than or equal, it must be less than -4. We rewrite the inequality as x < -4, and graph it relative to the domains given. Real numbers include all numbers, so we shade in the number line with a solid line. We use an open circle at -4 to show that x doesn t equal -4, but is close to it: b) Since whole numbers do not include negative numbers, there is nothing to graph. We call this an empty set, which is sometimes represented by a zero with a slash through it, like this: 0 Practice Set 1 A calculator is not necessary for any problems in Practice Set Match the definition of mathematics with its author. Choices include: Euler, Kline, Sawyer, Shormann, Whitehead A) The classification and study of all possible patterns =. B) A God-given tool for measuring and classifying patterns and shapes =. C) The science which investigates the means of measuring quantity =. D) A study of space and quantity =. E) The foundation of exact thought as applied to natural phenomena =.

9 2. Mathematics helps us know that if we search for, we can often find it. 3. God s divine nature is also, because He exists regardless of our feelings and beliefs about Him. 4. Even though it s true, nobody completely understands. 5. Understanding some things, but not everything, about a concept requires, patience, joy, perseverance and prayer. 6. Fill in the blanks in this table on different ways to describe unity and diversity. Unity Diversity God Fully Divine, Fully Human General Concrete Eternal Finite Continuous Many 7. John Saxon said A number is an. 8. True or False. All integers are real numbers.. 9. Natural numbers are also called numbers. 10. Convert CLXXIV to the Hindu-Arabic numeral system. 11. Convert CMXCV to the Hindu-Arabic numeral system. 12. Convert 48 to Roman Numerals. 13. Convert 547 to Roman Numerals. 14. Convert the binary number 111 to base Convert the binary number to base Convert the base 10 number 8 to base 2. Hint: 2 3 should be on, and everything else is off. 17. Convert the base 10 number 29 to base 2. Hint: 2 4, 2 3, 2 2 and 2 0 should be on, and everything else is off. 18. Graph x > 0, Domain = Counting Numbers. 19. Graph x > 0, Domain = Real Numbers. 20. Graph x < 5, Domain = Integers.