The Real Number System

Question 1

What is a set?

Answer 1

A collection of objects called the elements or members of the set.

Question 2

The elements of most sets are usually what?

Answer 2

Numbers

Question 3

These are used to enclose the elements of a set?

Answer 3

Set braces like { at the beginning and } at the end.

Question 4

Name one element of this set {1,2,3}?

Answer 4

2; other answers include 1 or 3

Question 5

What is the set N?

Answer 5

The set of natural numbers or counting numbers is an infinite set which only includes positive whole numbers beginning with the number 1.

Question 6

What is the set containing no elements called and what is its symbol?

Answer 6

The null set. Expressed as the number zero with a diagonal line through it. Or written as { }.

Question 7

Does { 0 with diagonal through it } equal the null set?

Answer 7

No. The null set has zero elements. This set still has one element.

Question 8

2 E {1,2,3} means what?

Answer 8

That 2 is an element of set {1,2,3}

Question 9

What does E with a diagonal line through it mean?

Answer 9

It means whatever is to the left of this expression is not an element of the set to the right of this expression.

Question 10

In Algebra, what represents numbers? Or what is used to define a set of numbers?

Answer 10

Variables. And the most commonly used variables are w, x, y, z.

Question 11

What does {x | x is a natural number between 3 and 15} represent?

Answer 11

This set builder notation which says that ‘the set of all elements of x is such that … in this case x is a natural number between 3 and 15 and that means the answer for this set is {4,5,6,7,8,9,10,11,12,13,14}.

Question 12

Once you learn to write sets using set notation, what is the next logical step to learn?

Answer 12

How to use this information visually using number lines, to view it and see it…differently for visual learners.

Question 13

After you learn what number lines mean, what should you know next in Algebra?

Answer 13

You should gain a broad understanding of all the different types of sets of numbers so you have a canvas to work with so to speak, to learn the rest of all the other Algebraic concepts that exist. To sum up all sets you have C for Complex. Under the set of Complex numbers you have all Real Numbers. All Real Numbers may also be expressed in Complex Notation. Real Numbers consist of Irrational and Rational Numbers. Irrational Numbers (S) can’t be expressed as Fractions, thus these are numbers that do not have repeating decimals, but these are non-repeating decimals which may be expressed infinitely, or more commonly with other math simplifying expressions like a square root symbol or pi. Rational Number are the set of repeating decimals and fractions p/q. Integers are whole numbers which may be negative, zero, or positive. Whole Numbers is the set of numbers including 0 and all other natural positive counting numbers. Interestingly of all the numbers that exist that do exist, they all consist of the first 10 elements of this set {0,1,2,3,4,5,6,7,8,9}. Last we know that the set of Natural Numbers is all counting numbers beginning with 1 that are non-zero and non-negative…{1,2,3,4,5,6,7,8,9,10,11,12, …}

Question 14

True or False: Variables represent numbers like pronouns represent persons, places, or objects.

Answer 14

True. Variables stand for numbers.

Question 15

What is the application of set builder notation?

Answer 15

To specify solutions of dependent equations. Ch. 4 is about the systems of equations where this is done.

Question 16

List the elements in {x | x is a natural number greater than 12}

Answer 16

{13,14,15,16,17,18,…}

Question 17

Listing the elements in a set is a skill that reinforces understanding of set values. List three elements in the C, R, S, Q, I, W, N sets.

Answer 17

0i + 2, 4i - 3, 5i + 1,000, (square root of -37); 34, pi, 5/6; square root of 2, pi, -(square root of 37); 5/11, 1/2, 4 (4/1); -4, 0, 4; 0, 5, 100 (no negatives); 1, 2, 100000000001

Question 18

True of False: There is more than one way to describe a set in set builder notation. If so give an example.

Answer 18

Yes. {z | z is a natural number greater than 6} will be the same set as {z | z is an integer greater than or equal to 7}. Both sets equal {7,8,9,10,…}. This is because there are sets within the set of integers that are also known as natural number sets. These two overlap.

Question 19

Can the variable used to describe a set be different? True or False.

Answer 19

True. Concept Check. A man claims {x | x is a natural number greater than 3} and {y | y is a natural number greater than 3} actually name the same set, even though different variables are used. Is this man correct? Yes because the set is the set no matter which variable defines it.

Question 20

What is used to get a good picture of a set of numbers?

Answer 20

A number line. Also for further explaining the properties or arithmetic like adding and subtracting, you may view these visually with number lines. Or as with counting using your hand to do basic arithmetic.

Question 21

What is the interesting thing about creating number lines that is similar to a real world application.

Answer 21

It’s a way of constructing something that never existed before. Same as in construction when a new structure, building, office, or home is built it all begins with an architect drawing lines and measuring them to certain dimensions on a set of blueprints. Number Lines allow you to begin a career in the field of architecture or construction. Because lines may be expressed in finite terms numerically or infinitely too. Or as infinite rays. Additionally, all sets of numbers but C are visible on number lines. Which begs the question how do we graph complex numbers or represent them visually also using Geometry. We will look at this later.

Question 22

What is the one set of numbers you can’t see on a number line?

Answer 22

You can’t see the complex set of numbers because this is an imaginary set that we will learn how to construct later. For now just know it is constructed as a + bi or a - bi for each complex number which is imaginary.

Question 23

Why is the set of R real numbers a subset of the Complex if Complex numbers are imaginary?

Answer 23

This is because Complex numbers are expressed partially and as the addition of a real number and an imaginary one. This means each complex number is also partially as an imaginary number but when the imaginary part is equal to zero which it may be at times, the real number remains, and also the means of expressing this even as an imaginary number still remains in tact too. You can therefore construct an imaginary number from a real one essentially. It’s important to remember this in math. Similar to construction you are able to build new concepts on basic foundations or tear down complex foundations back to basic ones in some instances. Mathematicians manipulate these foundations to make new discoveries. Calculating pi is one for of manipulation by using approximations and similar geometric shapes.

Question 24

If you are able to express numbers on a number line, what is this a further example of?

Answer 24

Graphing. And since graphing a line is the easiest graph to comprehend and create, so easy a cave man can do it, then everyone should have confidence in their graphing skills.

Question 25

What is a terminating decimal and what type of number does it represent what set of numbers?

Answer 25

A terminating decimal is a number like .125. This represents a rational number that may be expressed as 125 / 1000 and since the number is finite and expressed as a fraction we know that this number is a real rational number that is not an integer or whole number or natural number. A whole number like zero is a whole number, and an integer and a rational number because 0 / 1 = zero, and it is also real, and interestingly it is also Complex when a and b = zero. All whole numbers are complex because when a = whole number and b = zero, the result is equal to the whole number.

Question 26

What is a non-terminating decimal? And how many sub types of non-terminating decimals are there?

Answer 26

A non-terminating decimal is one that doesn’t have a finite end to it but nor does it necessarily repeat forever. It may repeat forever like 5/11 or it may be a decimal that does not repeat like pi but which also never ends. It’s important to distinguish between non-terminating and repeating because a non-terminating type of number may be irrational or rational. A non-repeating number may only be irrational. All repeating numbers are rational because of the nature of LONG DIVISION.

Question 27

Describe the concept of infinite numbers just looking at a number line and between two natural numbers which are whole numbers and integers. While the distance between a number labeled as 1 and a number labeled as 2 on your number line is just 1 unit, there’s actually an infinite number of numbers (or distances) between any two numbers which are different, and you may see this visually on a number line. How does this relate to geometry and the circle?

Answer 27

The Circumference C of a circle is 2 * pi * r. 2*r = the diameter d. Thus the circumference C by substitution is also expressed as pi * d, but the term of interest is pi because it is an irrational number that does not repeat itself at any point in time but IT ALSO does not terminate, BUT IT IS ALSO NOT IMAGINARY. Pi is a term used to express its meaning, but pi is not a variable. Pi is a way of remembering that irrational numbers are still classified as real numbers that represent real distances but when you think of pi as a line segment that is bent into a circle like a rope that is moved from a straight line into a circle the only thing to remember is that pi can be divided into an infinite number of smaller lines or segments the same as any other real number, but in a circle you express the distance of your rope as a measure of pi by referring to the real length of the diameter and pi to discuss the length of the circumference. THUS EVERY REAL NUMBER CORRESPONDS TO A POINT ON A NUMBER LINE. THIS BEGS A FURTHER QUESTION AS RELATED TO LONG DIVISION.

Question 28

This formula is based on the fact that the circumference of a circle is always equal to its diameter multiplied by pi.

Answer 28

This is the Liebniz series formula. pi = 4 * (1 - 1/3 +1/5 - 1/7 + 1/9…)

Question 29

Why should you understand the difference between repeating and non-repeating numbers, plus the difference between terminating and non-terminating numbers.

Answer 29

Because you can have non-terminating numbers that are rational and repeat forever like 5/11, or you can have non-terminating numbers that have no repeating pattern and these are call irrational numbers, such as the square root of 2, or most commonly pi which equals 3.14159….

Question 30

What is a repeating number classified as?

Answer 30

A Rational number.

Question 31

What is a non-repeating number classified as?

Answer 31

Either Terminating and non-repeating or not terminating and non-repeating. The first is rational. The second is irrational.

Question 32

Why is understanding how to calculate pi helpful for understanding the real number system.

Answer 32

Because if pi is an unknown number where there’s no set p and q that p/q = pi so this means pi is not A RATIONAL NUMBER, then pi is an approximation…and in order to find an approximation, we know we have to assume two numbers that can be divided by each other that should give us the value the of pi, and this is what the method of Archimedes did with polygon approximation and trigonometry functions. By comparing the values of inscribed polygons to circumscribed ones around a circle, the ratio of pi is the perimeter of the circumscribed polygon divided by the perimeter of the inscribed one. There’s 2 other methods to find the value of pi, one is by infinite series of Liebnitz. The other is the Monte Carlo method of using points inside a square inscribed in a circle. Curious how the Liebnitz formula was discovered however as feel in addition to Archimedes is easiest to understand.

Question 33

True or False: Once you identify examples of numbers sets you will have a better understanding of the relationships that exist between number sets.

Answer 33

True. And relationships between different types of numbers and how they are represented visually lead next into number properties such as inverse, absolute value, inequality relationships, interval notations, and the order of operations for real numbers; some properties further include commutative, associative, identity, distributive, multiply, divide, add, subtract. The additive inverse and multiplicative inverse are properties too. You can insert the additive inverse or the multiplicative inverse into numerical calculations when solving problems in the order of operations.

Question 34

Adding R (meaning real) numbers. To add two numbers with the same sign the result will also have the same sign after you take the ABSV of each number and then add them together. When the signs differ, take the ABSV of each number, subtract the smaller from the larger, and add the sign of the larger number to the answer if after this it differs. don’t change anything if the smaller ABSV has the opposite sign of your answer. So can you subtract with ABSV too?

Answer 34

Yes.

Question 35

Distance on a number line is a measure of ABSV. If you take the ABSV of 4 and the ABSV of 1 and then you subtract 1 from 4, you get the ABSV of 3 as your answer. So before you make your calculation determine if in a minus b or a - b if a is larger than b. if b is larger than a then take the ABSV of each number and subtract the answer of b from a to get a negative answer. is 5 subtracted from eighteen negative or positive.

Answer 35

positive.

Question 36

What is the additive inverse of a variable a. AND WHY IS IT SO FUNDAMENTALLY IMPORTANT.

Answer 36

The answer is -a. Since the numbers must add to 0. You can use this to switch between addition and subtraction. ALSO IT IS FOUNDATION FOR LEARNING COMMON ADDITION SOLUTIONS AND MULTIPLICATION FACTS AND TABLES. If you have a - b this now becomes a + (-b). If you have addition of a + b. This now becomes a - (-b). THE DIFFERENCE BETWEEN THE ADDITIVE INVERSE AND THE MULTIPLICATIVE INVERSE IS THAT FOR THE ADDITIVE PROPERTY ADDITION BECOMES SUBTRACTION TO 0, AND a must add with another number to zero and this number is the inverse of a with is -a, so by taking a + (-a), -a becomes the additive IDENTITY which when added to a equals 0. In Multiplication the magic number for a is 1. We are attempting to find the number when multiplied by a will equal 1 and this means you now have to divide a by itself and then multiply a by the result of 1 divided by a, and the equation becomes a * 1/a = 1. THIS MEANS THE TERM 1/a is extremely significant for algebraic equations for SOLVING THEM AND MANIPULATING THEM LATER WITH THE NUMBER PROPERTIES (COMMUTATIVE, ASSOCIATIVE, DISTRIBUTIVE.). Consider for the 10 numbers that make up all numbers, 1/a means a = 1 is the additive identity, however 1/2, 1/3,1/4, 1/5, 1/6, 1/7, 1/8, 1/9 all reveal some interesting facts about numbers and mainly of these numbers both 1/3 and 1/9 have repeating decimals but 1/9 = .1111111111… and this is the building block for all other numbers that do repeat ONE digit only indefinitely. 2/9 = 2222222, and 3/9 = .3333333 and 3/9 = 1/3 simplified. THE ONLY OTHER SIGNIFICANT DIVISOR FOR REPEATING DECIMALS IS GOING TO BE 11 because this divisor allows repeating 2 digit decimals AGAIN FOR MULTIPLES OF 9. 1/11 = .09090909, 2/11 = .18181818. (18 IS DIVISIBLE BY 9), 3/11 = .27272727 (27 IS DIVISIBLE BY 9), …, 10/11 = .909090 (90 IS DIVISIBLE BY 9), THEN 12/11 = 1 AND 1/11 = 1.0909090909 (SO THEY BEGIN TO REPEAT AGAIN IN MIXED NUMBERS AFTER THE DECIMAL POINT). IF YOU ADD 1/9 TO 4/9 THIS MEANS YOU GET 5/9 OR .5555555. YOU HAVE TO ADD 8/9 TO 1/9 TO EQUAL 1.

Question 37

What is the multiplicative inverse of a.

Answer 37

The answer is 1 / a. Since the numbers a and 1/a must multiply to 1. You can switch between division and multiplication with this too.

Question 38

Find the square root. The positive square root is the principal square root and uses the radical sign as the symbol for this. The negative square root of a number is written as -(square root of x) with the radical sign. The square root of a negative number is not a real number because to use the radical sign means the answer must be positive if there is an answer. This is where we first discover that through exponents we get to roots but in roots we see that some numbers do not exist or at least they do not exist in the real number system that is. You can take the square root of zero and that answer is also 0, but zero is also neither positive or negative. Is exponential arithmetic apart of the order of operations?

Answer 38

Yes. After solving for parenthesis and working within the brackets or ABSV bars, start with the innermost set and work outwards. Then when you evaluate powers, roots, and ABSV, you can then move into M/D next.

Question 39

What is Please Excuse My Dear Aunt Sally?

Answer 39

The order of operations. Parenthesis. Inner set. Exponents. Multiply. Divide. Add. Subtract.

Question 40

In PEMDAS are MD done in the order in which they appear?

Answer 40

Yes.

Question 41

Algebraic expression can includes any of the letters in the alphabet as variables? T or F

Answer 41

T

Question 42

The distributive property provides a way to write a product as a sum.

Answer 42

True

Question 43

Parenthesis used on one side of the distributive property are removed on the other side …with the ) symbol, true or false?

Answer 43

True

Question 44

For the multiplicative inverse property a can be zero, true or false?

Answer 44

False because you cannot divide 1 by 0.

Question 45

Each of addition and multiplication has an identity property. a + 0 and a * 1? True or False.

Answer 45

True.

Question 46

Terms are only interchanged with the commutative property not the associate property. The only thing that shifts with the associative property is the parentheses but the order of terms remains the same. T or F.

Answer 46

T.

Question 47

Using the properties of numbers is like using the PEMDAS in reverse. BECAUSE YOU CAN THINK OF SOLVING THE EQUATION BY ADDING PARENTHESIS THAT MAY NOT BE THERE, THEN COMBINING LIKE TERMS SO THE EQUATION -2m + 5m + 3 - 6m + 8, can be rewritten as (-2 + 5 - 6)m + 3 + 8 which simplifies to -3m + 11. Or you can take the long route and go (-2 + 5)m + 3 - 6m + 8, then simply to 3m + 3 - 6m + 8, then simplify to (3 + -6)m + 3 + 8 to arrive there. One goal is to isolate your variable from the other terms to solve these equations.

Answer 47

True

Question 48

a * 0 = 0 and 0 * a = 0. This is the multiplication property of 0, true or false?

Answer 48

True

Question 49

Is the exact value of the number pi known?

Answer 49

No because the number is a non-repeating and non-terminating decimal CLASSIFIED AS A REAL NUMBER WHICH IT IS. I read that in 2019 a supercomputer was used to calculate the value of Pi to 62.9 million digits. But similar to how dividing 5 by 11, you get an infinite answer, that answer is similar to the value of Pi which has no end. As of now we know the number to be the ratio of a circle’s Circumference to it’s diameter or C/d = Pi which is ~3.141592653589793… which is as far as most calculators care to take it.

Question 50

Square Roots lead us to discover what about the broad set of all number sets?

Answer 50

That since you cannot take the square root of a negative number, any number that is negative that is raised to 1/nth power where n is an even natural number greater than zero, does not exist. THIS MEANS THROUGH ROOTS WE HAVE DISCOVERED A WAY TO EXPRESS NUMBERS THAT DO NOT EXIST.

Question 51

WHY IS 1/a an important concept to understand?

Answer 51

just using {0,1,2,3,4,5,6,7,8,9,10,…} you can see what it takes for a number to repeat itself but also never terminate. The number 1 is NOT PRIME. 2 IS THE FIRST PRIME NUMBER. ALSO THE NUMBER 9 IS NOT PRIME, BUT YET 1/9 = .111111…, 2/9 = .2222222…, 3/9 = .333333…, 4/9 = .4444444…, 5/9 = .5555555…, UP TO 8/9 = .88888888…, and 9/9 = 1.0, but 10/9 = 1 and 1/9 = 1.11111, 11/9 = 1 and 2/9 or 1.22222, etc., and 19/9 = 2 and 1/9 or 2.11111, means 28/9 = 3.11111, 37/9 = 4.11111 (adding the 10 from 10/9 plus nine in the top numerator equals 19/9 plus repeating this and adding another 9 in the numerator is = 28/9 , then 37/9, repeated….we start to set series develop here too. SO HOW DO YOU GET .10101010101, AND THEN .1212121212, AND THEN .1313131313 TO GET THE PATTERN GOING FOR ALL NUMBERS REPEATING THEMSELVES, YOU JUST. 11 IS THE MAGIC NUMBER FOR A DIVISOR FOR REPEATING ANY SINGLE # INDEFINITELY. NOTICE HOW 12/100 = .12 IS NOT THE SAME FRACTION AS .1212 = 10908/90000 WHICH SIMPLIFIES TO 301/2500 SO EACH DECIMAL TO THE EXACT DECIMAL HAS A FRACTION TO THAT THAT EXACT DECIMAL WHEN THE NUMBER IS NOT WRITTEN AS AN INFINITE NUMBER. ANY # DIVIDED BY 9 HAS A REPEATING DECIMAL THAT IS THE SAME WHETHER THE NUMBER IS PRIME OR NOT AND NOT EQUAL TO 9. ANY # DIVIDED BY 11 NOT EQUAL TO 11 HAS A 2 DIGIT REPEATING DECIMAL.

Question 52

How do you know which inequality is less than and which is greater than?

Answer 52

The pointy side of each inequality symbol points to the lesser number but for the “LESS THAN” meaning this point is found on the left side <. For the greater than symbol this is found on the right side >. There’s also interval notation to express sets on number lines using the following: ( and ]. and [ and ); as an example (2, 13] is an open interval at 2 but does not include 2 and is a closed interval at 13 and does include 13 per these symbols.

Question 53

What is the closure property?

Answer 53

The closure property applies for real numbers under addition, subtraction, multiplication, and division if you add, subtract, multiply, or divide two real numbers, the result will always be a real number. Since Real numbers include both irrational and rational numbers it means that number like the square root of 2 times the square root of 6 or the square root of 12 when simplified still each another real number, in this case 2 times the square root of 3.

Question 54

What is the commutative property and why is it important?

Answer 54

Commutative property: The order in which you add or multiply real numbers does not matter. For example, 3 + 4 is the same as 4 + 3, and 3 * 4 is the same as 4 * 3. NOW ONCE YOU BEGIN TO SOLVE EQUATIONS YOU WILL LEARN THAT IT IS OKAY “TO MOVE” THE NUMBERS IN SUCH A WAY THAT IT MAKES THE EQUATION EASIER TO SOLVE…AS YOU USE OTHER PROPERTIES LIKE ASSOCIATIVE AND DISTRIBUTIVE TO ARRIVE AT ANSWERS. THINK OF THE WORD COMMUTE WITHIN COMMUTATIVE WHICH MEANS “TO MOVE OR TRANSPORT”. YOU MAY MOVE NUMBERS JUST AS EASILY AS YOU MAY MOVE ALGEBRAIC TERMS REPRESENTED BY LETTERS in equations. (w, x, y, z). Ex. 3x + 4y -9 = 4y +3x -9.

Question 55

Describe the associative property?

Answer 55

Associative property: The grouping of real numbers does not matter when you add or multiply them. For example, (3 + 4) + 5 is the same as 3 + (4 + 5), and (3 * 4) * 5 is the same as 3 * (4 * 5). This also works with more than just 3 numbers as 1* (234) = (123) *4. It allows us to simplify complex mathematical expressions and make them easier to solve.

Question 56

Why is this questions about properties so tricky as a word problem: Bob needed to multiply 6 by 5, then multiply that answer by 4. He decided to first multiply 5 and 4 to get 20. Then bob multiplied 20 by 6 to get 120. Which one of the following is valid concerning the steps Bob took to solve this? a) his answer is wrong because he should have first multiplied 6 by 5. b) his answer is wrong because he did follow a correct sequence of steps. c) his answer is right and he used the associative property of multiplication. d) his answer is right and he used the commutative property of multiplication.

Answer 56

The way the problem reads it sounds like we have 6 * 5 * 4 = 5 * 4 *6. By first glance it appears that the order of the numbers has been moved similar to how you see the commutative property. ….because like that property states the order in which you add or multiply numbers does not matter per this property. INSTEAD IT MEANS THAT (6 * 5) * 4 = (5 * 4) * 6 BECAUSE TO PROPERLY SOLVE THE PROBLEM YOU HAVE TO INCLUDE THE PARENTHESIS FOR THE INDIVIDUAL CALCULATIONS FIRST WHICH ARE STEPS YOU TAKE IN ROUTE TO THE SOLUTION, MEANING YOU ARE USING THE ASSOCIATIVE PROPERTY OF MULTIPLICATION BECAUSE ANYTIME TWO NUMBERS INTERACT THEY THEN FORM A “GROUP” THAT MAY BE ENCLOSED BY PARENTHESIS. BASED ON HOW THE PROBLEM WAS ACTUALLY SOLVED YOU NOTICE THE NUMBERS ARE JUST INTERCHANGED, AND SINCE WE HAVE 3 NUMBERS THIS PROPERTY APPLIES INSTEAD OF THE COMMUTATIVE.