1.1 Real Numbers: Algebra Essentials

Summary

In this introductory algebra lesson from Ryan Melton, foundational concepts of real numbers are explored. It begins with an explanation of number sets including natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Understanding how to identify and work with these numbers is essential for interpreting decimals and performing arithmetic operations using properties like commutative, associative, distributive, identity, and inverse. The lesson also discusses ordering operations (PEMDAS) for evaluating expressions, explains the difference between equations and formulas, and guides on simplifying expressions and determining whether they are rational or irrational.

Highlights

• The lesson begins with a breakdown of number sets: natural, whole, integers, rational, and irrational. 🧠
• Ryan explains the termination and repetition of decimals for rational numbers. 💡
• Real number properties like commutative and distributive help simplify expressions and handle numbers efficiently. 🤓
• Examples showcase the practical application of PEMDAS in evaluating expressions. ➕
• The difference between a formula and an equation is clarified, with examples for hands-on understanding. 📚

Key Takeaways

• Understanding different sets of numbers is crucial! Natural numbers start from 1, whole numbers include 0, integers add negatives, rational numbers are fractions, and irrational numbers don't end. 🔢
• Rational numbers can be terminated or repeated when expressed in decimals, whereas irrational numbers cannot. 🌀
• The properties of real numbers - commutative, associative, and distributive - are your best friends for algebra! 🤝
• Use PEMDAS to solve expressions in the correct order: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). 🧮
• Formulas are more general than equations and can involve multiple variables. Be sure you can distinguish between them! 📏

Overview

In Ryan Melton's enlightening session, learners are introduced to the foundational world of real numbers, covering everything from counting numbers to complex fractions and square roots. The lesson delves into differentiating natural numbers from integers, and rational numbers from their unpredictable partners, the irrationals.

Ryan brings clarity to decimals, demonstrating how rational numbers can be neatly terminated or repetitively cyclic, unlike the endless swirl of irrational numbers. He guides students in identifying the nature of numbers, using examples like square roots and fractions, to determine their status and behaviors.

Equipped with the properties of real numbers and the organizational tool of PEMDAS, attendees learn to tackle complex algebraic expressions with confidence. The lesson concludes with distinguishing formulas and equations, ensuring that the burgeoning mathematician knows how to identify and use each efficiently.

Chapters

• 00:00 - 03:00: Introduction to Sets of Numbers The chapter introduces sets of numbers essential to algebra, beginning with natural numbers used for counting like 1, 2, 3, and so on. It then describes whole numbers, which include all natural numbers and zero, such as 0, 1, 2, 3, etc. The set of integers is explained as a combination of whole numbers and their negative counterparts, extending indefinitely in both directions on the number line.
• 03:00 - 06:30: Rational and Irrational Numbers This chapter explores different types of numbers, starting with integers which include both positive and negative whole numbers, as well as zero. It then moves on to rational numbers, which are expressed as fractions where the numerator and denominator are integers, and the denominator is not zero. Rational numbers can also be represented as repeating or non-terminating decimals.
• 06:30 - 11:00: Classifying Numbers The chapter titled 'Classifying Numbers' focuses on the different types of numbers, specifically fractions and irrational numbers. It explains that fractions can either be terminating (like 0.5) or repeating (like 3.3). It then defines irrational numbers as those that cannot be expressed as fractions, hence they neither terminate nor repeat. An example problem is presented, illustrating how to write different numbers as rational numbers, such as expressing '7' as '7/1' and '0' as '0/1'.
• 11:00 - 16:00: Order of Operations and Grouping Symbols This chapter focuses on understanding and applying the order of operations along with using grouping symbols in mathematical expressions. It emphasizes the importance of carrying out operations in the correct sequence to achieve accurate results. The chapter includes examples and explanations on converting rational numbers into decimal form through the process of long division, identifying them as either terminating or repeating decimals. A practical example provided includes simplifying the division of -8 by 1 and converting rational numbers to decimals by dividing the numerator by the denominator, specifically highlighting the pattern recognition in decimals whether they terminate or repeat.
• 16:00 - 23:00: Properties of Real Numbers The chapter focuses on the properties of real numbers, highlighting concepts such as rational numbers and their characteristics. Specifically, it addresses repeating decimals and terminating decimals, with examples for clarity. For instance, it discusses how 15 divided by 5 equals 3, a terminating decimal, and illustrates repeating decimals, demonstrating the concept with calculations like 13 divided by 25 equating to 0.52.
• 23:00 - 28:00: Simplifying Expressions Using Properties The chapter titled 'Simplifying Expressions Using Properties' covers the process of determining whether numbers are rational or irrational. It provides methods to identify if rational numbers are terminating or repeating decimals. An example is given to illustrate how multiplying the numerator in a fraction influences its decimal representation.
• 28:00 - 35:00: Evaluating Expressions with Given Values The chapter 'Evaluating Expressions with Given Values' begins with exploring the concept of rational numbers using the examples of the square root of 25 and the fraction 33/9. The square root of 25 is identified as 5, which can be expressed as 5/1, categorizing it as a rational number. This is further confirmed by recognizing 5.0 as a terminating decimal. Similarly, the fraction 33/9 simplifies to 11/3 and can be expressed as 3.6 repeating. These examples illustrate the properties of rational numbers, including their ability to be expressed as fractions and identified as repeating or terminating decimals.
• 35:00 - 42:00: Formulas and Equations The chapter titled 'Formulas and Equations' discusses the characteristics of square roots and rational numbers. It explains that if a square root, such as the square root of 11, cannot be simplified to an integer or a rational number directly, it is considered irrational. This means it does not terminate or repeat. Additionally, the transcript includes a brief explanation of simplifying fractions, using 17 over 34 simplifying to 1/2 as an example.
• 42:00 - 47:00: Simplifying Algebraic Expressions The chapter focuses on simplifying algebraic expressions, particularly examining whether given numbers are rational. Several examples are provided, such as the number 0.5, which is rational and terminating. In contrast, a more complex sequence like 0.303330333033 cannot be classified as rational because it does not terminate or repeat. The chapter concludes with a reflection on distinguishing between rational and irrational numbers.
• 47:00 - 50:00: Conclusion and Recap of Properties In the conclusion and recap chapter, the focus is on the properties of rational and irrational numbers. The discussion emphasizes the example of the square root of 11 and how it cannot be expressed as a fraction of two integers a/b. By equating the square root of 11 to a/b, it's shown that 11 equals a squared divided by b squared, which leads to the equation a squared equals 11 times b squared. The chapter encourages proving the irrationality by attempting to find integer solutions for a and b, which in practice do not exist, thus concluding that the square root of 11 is irrational.

1.1 Real Numbers: Algebra Essentials Transcription

• 00:00 - 00:30 section 1.1 real numbers algebra essentials here right so there are some sets of numbers that we deal with most commonly the first is the set of natural numbers these are numbers that we use for counting get 1 2 3 etc then there's the whole numbers which is the natural numbers and we include 0 so 0 1 2 3 etc the set of integers combines the negative natural numbers with the set of whole numbers so we have beginning at negative infinity as a direction okay a
• 00:30 - 01:00 negative 3 negative 2 negative 1 0 1 2 3 etc we continue in both directions and we have the opposites of all of our natural numbers in them well then we have the rational numbers which are fractions written as M divided by n where m and n are integers and in our denominator is not zero every number that is a rational number is a repeating or a non terminating decimal okay as if
• 01:00 - 01:30 fractions are either terminating as in they stop to point 5 or they repeat three point three repeating something like that the set of irrational numbers are the set of numbers that are not rational so therefore they are not repeating and their non terminating alright so example one write each of the following is a rational number the simplest way to write each of these as a rational number is for Part A 7 divided by 1 0 is 0 divided by 1 and C is
• 01:30 - 02:00 negative 8 divided by 1 that is the simplest way although there are others all right now write each of the following rational numbers as either a terminating or repeating decimal now the way we will do this is we will divide the numerator by the denominator and using our long division if we see that something is there's a pattern emerging something that's repeating or it stops we have no remainders at some point so I'm going to cut to the chase here this first one is negative zero point 7 1 4 2
• 02:00 - 02:30 8 5 and at that point it begins repeating so this one is repeating and therefore of course it is a rational number 15 divided by 3 15 divided by 5 is 3 and therefore that is a terminating decimal because I could write this as 3.0 if I need a decimal expansion there all right 13 divided about 25 that is 0.5 2 which
• 02:30 - 03:00 you can see if you multiply the numerator and a linear by 4 very quickly 0.5 2 so that is a terminating decimal determine whether each of the following numbers is rational or irrational if it's rational determine whether it's terminating or repeating so let's go ahead and start I'm going to write out a chart here determining if they are rational or not if they are terminating or if they are repeating well the square
• 03:00 - 03:30 root of 25 is 5 since we can write that as 5 over 1 that is rational and because it's 5.0 that is also a terminating decimal all right 33 ninths that is equivalent to 11 thirds which is 3.6 repeating so it's rational because I can write it as 11 thirds and it is a repeating decimal because of the 0.6 repeating Khayyam part c square root of
• 03:30 - 04:00 11 it's a it's a very common argument that we could pose here but I'm just going to just go ahead and say if it is a square root that does not simplify to an integer or a rational number directly the square root 11 doesn't simplify this is going to be irrational it's not rational therefore it doesn't terminate or repeat 17 over 34 simplifies to 1/2
• 04:00 - 04:30 or 0.5 so it is rational and terminating now part e zero point three zero three three zero three three three zero three three three three probably a zero there doesn't terminate and doesn't repeat so therefore it must be not rational whoops it should be not rational now add some food for thought
• 04:30 - 05:00 I'm just going to throw this out this is this extra this is not required if square root of 11 were rational then it would be written as a divided by B which means 11 equals a squared over B squared okay which means that a squared equals 11 B squared if you can find a solution to that equation to values a and B that make this true and I should go ahead and
• 05:00 - 05:30 note with B naught 0 then you could say it's rational but the fact is there is there are no numbers a and B that'll make that true so we have to just sort of wave our hands at this point and go well square root of 11 must not be rational just a thought there number 4 classify each number as either positive or negative and as rational or irrational is it on the left or the right of zero on the number line
• 05:30 - 06:00 okay we can tell because negative numbers always have a minus or a negative sign preceding them all right so this is negative I'm just going to go ahead and write these the second one is positive next is negative the next is also negative and the last is positive and that's the first thing all right so that means that just go ahead and put zero down here our negative numbers go
• 06:00 - 06:30 on the left our positive numbers go on the right so negative 10 thirds will be on the left the square root of five will be on the right see the negative square root of 289 is negative so it will be on the Left D negative six pi will be on the left as well because it's negative and E is positive so it will be on the right now as far as rational or irrational a is clearly rational just go
• 06:30 - 07:00 ahead make label that with an R square root of five is irrational for the reasons we mentioned that last question- square root of 289 that is negative 17 negative 17 over 1 so that would be rational negative 6 pi is irrational there's a much different argument to show that's irrational but we'll leave that for another day and Part II if you notice 6 1 5 3 8 4 at
• 07:00 - 07:30 that point that begins repeating so this one is rational as well 5 classify each number as being a natural number a whole number an integer a rational number and or an irrational number so let's go ahead and leave make a chart here natural whole number integer which often
• 07:30 - 08:00 we use as e for that rational and irrational the apostrophe there to indicate that it's not the complement right square root of 36 is 6 and so therefore that is a natural number which is also a whole number it's certainly an integer and it's rational 8/3 that is a rational number okay we can say that is
• 08:00 - 08:30 rational it's not an integer it's not a whole number I'm not a natural number and since it's rational it can't be irrational so that's all we get there square root of 7 that is irrational that's all we have there so we have a choice of either rational or irrational that's the only option there but then once something is rational it can also be some of these other things before you see them alright negative 6 is not a natural number it's not a whole number it is an integer and therefore it's also
• 08:30 - 09:00 rational and then Part III point two one two one one two one one one two there's no repeating there's no terminating so it can't be rational it must be irrational next we have some operations the order that we evaluate a mathematical expression we use PEMDAS okay parentheses exponents multiplication division are on the same level in addition and subtraction these go left
• 09:00 - 09:30 to right we always go left to right when we are figuring multiplication division versus addition subtraction okay now there are lots of other acronyms other than PEMDAS that's just what we're going to stick with for the time being all right so we'll work inside our parentheses first so here that is going to be 6 squared minus 4 times 8 so we get 36 minus 32 and so this simplifies to for Part B
• 09:30 - 10:00 we'll work inside of parentheses which really that's a grouping symbols okay if something is a group together so our square root here the square root of 11 minus 9 or 2 is 9 so we can do that it's grouped together now on the left we have 25 minus 4 over 7 25 minus 4 is 21 over 7 so that would be 3 minus the square
• 10:00 - 10:30 root of 9 is 3 and so this simplifies to be 0 ok now I emphasize the grouping symbols idea because on Part C the absolute value of 5 minus 8 okay that is all grouped together so for Part C we have 6 minus the absolute value of and that would be negative 3 plus 3 times 3
• 10:30 - 11:00 subtracting those two the absolute value of negative 3 is positive 3 so this is 6 minus 3 plus I'll go and multiple that to be 9 now we're down to addition addition and subtraction we have no more grouping symbols so we'll go left to right 6 minus 3 is 3 plus 9 and then we'll add those two to get an answer of 12 right now for Part D we're going to simplify the numerator and the denominator separately but we will eventually divide that'll be
• 11:00 - 11:30 our last thing so in essence what we have is that these are grouped together our numerator and our denominator are grouped so for the numerator we have 14 minus 6 for our denominator we have 10 minus 9 that is a minus not a a multiplication there we go it's our numerator we have 8 divided by
• 11:30 - 12:00 1 and so our answer is 8 Part II of this beginning on the innermost part of this now if you notice we have brackets and parentheses this is again why I emphasize grouping symbols we will actually start in here and eventually get out here so for the first one that is 7 times 15 minus 2 bracket and I'm
• 12:00 - 12:30 going to simplify the 6-3 first that's 3 minus 4 squared plus 1 working our way inside there this is going to be 3 minus 16 plus 1 okay so we have 7 times 15 minus 2 times negative 13 so 7 times 15
• 12:30 - 13:00 is 105 plus 26 plus 1 and now we're down to addition so we can just go left to right 105 plus 26 is 131 plus 1 in our answer is 132 ok now the next thing we have to deal with is are the properties
• 13:00 - 13:30 of real numbers so this whole time we've actually been talking about one number system and that is real numbers and real numbers are generally indicated with a dub struck R&R with two lines on that vertical bar and the real numbers are our rational numbers are rational numbers combined with our irrational numbers anything anything that is either
• 13:30 - 14:00 rational or irrational follows all these same properties that we have listed here so we have first the commutative property which says that we can exchange the order with addition or multiplication we the associative property that says no matter how we group things under addition or multiplication things will work out they'll be the same distributive property says we can if we have multiplication along with our grouping we can do the multiplication first we can distribute we can spread
• 14:00 - 14:30 that our number around and that works well it's actually the same property because they it's it's with addition over multiplication so or multiplication over addition the identity property which says there is a number called 0 then we normally write a 0 such that if we can add it to any number a and it doesn't change anything we have the multiplicative identity which is any number x 1 x 1 is going to be that same
• 14:30 - 15:00 number we have the additive inverse or the opposite there are two numbers that one is positive one is negative if we combine them we add them we get zero and the multiplicative inverse which is any number except for zero every non-zero number every number except for zero has a multiplicative inverse and number we can multiply by it to get one and those
• 15:00 - 15:30 are both related to the identity property very much so so let's going we're going to go ahead and use these properties to simplify these expressions and I'm going to state what properties I use however I'm not going to write them okay so you can always come back this if you want to hear that 3 times 6 plus 3 times 4 well the distributive property says what I can actually do is add those to first and then multiply by three so that'd be 3 times 10 which is 30 Part B
• 15:30 - 16:00 I can change the way I group this according to the associative property I can say this is 5 plus 8 plus negative 8 8 and negative 8 are opposites so by our inverse property that is 5 plus 0 and then by our additive identity that is going to be 5 Part C Part C I'm not
• 16:00 - 16:30 going to really use any of my properties here and I will in a second all right so 6 minus we want to simplify the parentheses next that's 24 I'm going to leave that in parenthesis now our distributive property says we can change this to addition multiply our negative 1
• 16:30 - 17:00 so this is 6 plus negative 24 which is negative 18 Part D we can use several of our properties first thing we can do is change the order okay we can use the commutative property 4/7 times I'm going to rearrange 2/3 + 4 + 7/4 so the my commutative property by the associative
• 17:00 - 17:30 property I can regroup okay by the multiplicative inverse property 4/7 and 7/4 are inverses of each other so that is going to be 1 times 2/3 and then by the multiplicative identity that is 2/3 part E the distributive property says I can
• 17:30 - 18:00 multiply each of these by 100 which would be 75 minus R plus a negative 238 and adding those two together you get negative 163 now for number eight we have listing the constants and variables for each of these algebraic expressions constants are values that do not change so in this case and my constants here in
• 18:00 - 18:30 my variables here in this first one x + 5 5 does not change however we do not know the value of x so it can change in this case Part B 4/3 PI R cubed 4/3 and PI are both constants they are fixed whereas R is our variable Part C we have
• 18:30 - 19:00 2 is a constant we have several variables we have M and in here so those that are our constants and our variables number 9 evaluate this expression for each value of x have expression 2x - 7 so to evaluate that at 0 it is 2 times 0 - 7 2 times 0 is 0 - 7 so that is negative 7 to evaluate this at 1 that's
• 19:00 - 19:30 2 times 1 minus 7 it's going to be 2 - 7 or negative 5 its similarly for these next 2 2 times 1/2 - 7 2 times 1/2 is 1 that's our multiplicative identity and the inverse properties working there - 7 so we get negative 6 and evaluating at negative 4 2 times negative 4 minus 7 so that is negative 8 minus 7 which will be
• 19:30 - 20:00 negative 15 some more evaluating for example 10 evaluate x plus 5 for x equals negative 5 that would be negative 5 plus 5 we're evaluating value at x equals negative 5 well that is our additive inverse property so we get 0 there evaluate T
• 20:00 - 20:30 over 2t minus 140 equals 10 I'm going to plug value of 10 in 2 times 10 minus 1 so I get 10 divided by that's 20 minus 1 or 19 so 10 19th see R is 5 so 4/3 PI 5 cubed that is 125 so 4/3 PI times 125
• 20:30 - 21:00 125 times 4 would be 500 so 500 thirds PI Part D replacing a with 11 and B with negative 8 so 11 plus 11 times negative 8 plus negative 8 and all of that is not
• 21:00 - 21:30 necessary the way I'm writing all these parentheses but there are places where it is necessary so I go ahead and put them anyways now I can drop a couple of these this is B 11 and we minus 88 minus 8 and then simplifying the left to right that'll be negative 77 minus 8 or negative 85 part-ii replacing em with 2
• 21:30 - 22:00 and in with negative 3 this is the square root of 2 times M that's 2 cubed times N squared and here's a place where those parentheses are rather important alright so we know that 2 cubed is 8 so 2 times 8 and the negative 3 squared is positive 9 2 times 8 16 16
• 22:00 - 22:30 times 9 is 144 and so this simplifies to be 12 next thing we have is a formula versus an equation an equation is a mathematical statement indicating that two expressions are equal a formula is there an equation expressing a relationship between a constant and rate and some variable quantities so we are
• 22:30 - 23:00 going to have multiple variables in a formula versus an equation equation we might only have one but an equation is really often a specific kind of formula or a formula as a type of equation so these two are certainly related right so example 11 a right circular cylinder with radius R and height H has the surface area s in square units given by the formula s equals to PI R parentheses
• 23:00 - 23:30 R plus h close parentheses find the surface area of a cylinder with radius 6 inches and a height of 9 inches leave the answer in terms of pi right so our radius is 6 so R equals 6 and our height is 9 so this formula s equals 2 pi R parenthesis R plus h so if our example here we have 2 pi R so 2 pi times 6
• 23:30 - 24:00 parentheses I'm going to go and put all this in parenthesis parenthesis 6 parenthesis 9 now let's unpack that inside our larger parentheses we have 15 6 and we have 2 pi so multiplying 2 times 6 times 15 that would be 12 times 15 which is 180
• 24:00 - 24:30 so 180 PI square inches because that is the unit we were given something there is our surface here 180 PI inches squared now for example 12 simplify each of these expressions now we have some variables floating around here but a lot of things are going to work just like we
• 24:30 - 25:00 think they should so we want to recognize are things that are alike terms okay so I see that we have a 3x and we have an X here well the coefficient on the X term is a 1 so what we're going to do is we're going to take 3 plus 1 that'll be how many X's we have plus I've got this negative 2y minus 3y so
• 25:00 - 25:30 negative 2 minus 3 or negative 2 plus a negative 3y minus 7 so we're going to have 4x minus 5y minus 7 I'm running this in a little more detail than I need to but I do suggest detail as a rule Part B first we will distribute so we'll
• 25:30 - 26:00 remain with the two are distribute our negative five seven minus 15 plus 5 R plus 4 so the terms we can combine our 2 plus 5 are and then we've got plus our negative 15 plus 4 so 7r and that will be negative 11 or minus 11 there Part C
• 26:00 - 26:30 first distribute 40 minus 5s over 4 minus 2t over 3 minus 2's distributing that minus sign now the two terms we can put together first ok is going to be 4 minus 2/3 T and then we can also put together our negative 5/4 minus two S's
• 26:30 - 27:00 so 4 minus 2/3 is going to be 10 thirds 10 thirds T and that will be negative 13 yes negative thirteen fourths s negative 5/4 minus 8/4 will give us a negative 13
• 27:00 - 27:30 there Part D there are some terms we can put together you've got the M in terms so 2 + 3 MN - 5 mm + n those don't have any other components we can add into them so 5 m n minus 5m + n and that is all for that question number 13 simplify
• 27:30 - 28:00 this expression so we have a rectangle with a length of L and a width of W and it has a perimeter of P which is length plus width plus length plus width first we can combine those two and I'm going to go and skip the one step and say this is 2l plus 2w well based on our distributive property we can actually write P equals 2l + W that is actually the most simplified form of the
• 28:00 - 28:30 perimeter formula for a rectangle all right that brings us to the end of this section so remember all of those properties we listed we have the commutative associative the additive identity the multiplicative identity the inverses for each of those and we've also got orders for your property those come in handy when we were simplifying some fairly complex expressions