Hi, I'm Miriam. >> And, I'm Julie and we're your instructors for College Algebra, Animals, Architecture and Innovation. >> You'll be seeing a lot of our hands throughout the course, but we thought we would show you our faces too, so you could get to know us better. >> For you people out there, who don't think you are math people, we're out here to prove you wrong. Recent research has shown that students who have difficulties learning mathematics, have difficulties not because they're not smart, but because often they've had bad experiences learning mathematics. What's even better is that you can help your brain grow through doing math. This course is designed to help you play around with college algebra. Try to step out, make mistakes, try again, just get involved with it. Research also shows that making mistakes helps with learning. You're hear to learn about college algebra. We don't expect you to know everything. And you're not alone either, we have thousands of students all over the world taking this course. Talking about math can actually help you learn it better and the forum is a great place to do this. You can ask questions about things you don't understand or even just about cool algebra related things. So that's probably pretty different from other math classes you've taken before. It's theme based and it focuses on inquiry learning which means that we help you discover the mathematics on your own and show how it's relevant for your life. We hope you enjoy the course. Let's get started.
Since we're going to be doing a lot of math together in this course, a great place to start would be by talking about numbers, since we'll use them all the time. What is a number really? Well, it's not a thing out here in the world that we can see and touch. I at least have never seen a three or a negative 15 and a half walking around anywhere. Instead, numbers are abstract and we use them as tools to talk about things that we find around us. A kindergarten teacher, for example, needs to make sure that none of her students gets lost when she takes them on a trip to the zoo. One way she could do this would be by constantly looking at a list of their names and double-checking that every child on the list is in the group, but this would get really tiring and probably wouldn't be all that effective. What do you think might be a better way of doing this? Our first quiz of the course will be a really kind of silly question, but, let's go with it. How could a teacher make sure that none of her students get lost at the zoo? And here are a couple of answers that you get to choose from. Tell them all to be responsible for themselves and not run off. Your second choice is hope for the best. Your third choice is count them. Your last choice is they probably shouldn't go on the trip at, at all, just to be safe. Please click the circle next the choice below, that you think is the best answer. When you've picked which one you think is right, press the Submit button underneath.
Now this is kind of a funny question with some funny answers, and so depending on your thinking, you might picked any of these answers. I personally though, think that the best answer is the third one, to count them. If the teacher counts her students before they leave and then counts them at different points during their time at the zoo, she can know pretty easily whether or not they are all sticking together. Congratulations on completing your first quiz of the class whether you picked the same answer as I did or not. You'll encounter many quizzes like this throughout the rest of the course. Although they probably won't all be as silly as this one. But please remember that even as we dig deeper into algebra, the point of all of these quizzes will be for you to check if you understand the material. Just give each question some thought and if you're have trouble choosing an answer, rewatch the video or couple of videos preceding the quiz. If you get a question right that's awesome and if you miss one that's not a big deal at all. Just try it again and submit your new answer. You can redo each question as many times as you need to. We want you to do whatever you need to do to really understand the material.
So counting lets us think about quantities of objects and answer questions about them, whether it's how many students are in a teacher's class or how many moons orbit Saturn. Now, usually, when we count, we start with the number one and then we get to two, and three, four, and so on so forth. And we could keep counting like that until we get to huge numbers, like then, we could still keep going. Now, this special set of numbers has a name, they are called the natural numbers. So the natural numbers start with one and increase by ones forever. So now we want to make sure that you're totally solid on what falls under this category. Which of these numbers are natural numbers, 735, 6, slightly different kind of than then the last one. This time I want you to check off all of the answers that you think are right. So that might mean checking more than one, in some cases, you might even check off all of the answers that are there.
Remember that the only numbers that are natural numbers are 1, 2, 3, 4 and so on going up, but counting by ones. If we did that we would eventually get to 735 so this is a natural number. So is 6. 89.1 lies between two natural numbers. 89 and number because of the decimal of .1. -1000 and 1/3 are both less than 1, which is the smallest natural number, so neither of them belong in the set of natural numbers. Our last answer, 19 however is a natural number. We can already see this from the quiz, that there are a lot of numbers out there that don't qualify as natural numbers and we'll talk more about them very soon.
Now, let's think of another thing that we can count. How about something simple like cupcakes. Let's say that one of your friends bakes a bunch of cupcakes for your birthday party. He bakes 50 cupcakes. You are a very popular person. So a ton of people come and the cupcakes are really delicious, so pretty much everyone eats at least one. Halfway through the party people start coming up to you and asking you how many cupcakes are left. You look around and realize that you can't seem to find any cupcakes anywhere. They must have all been eaten. Can you tell your friends that there aren't any cupcakes left using a natural number? Please pick yes or no.
And the answer to this question is, no.
What number do you need to answer their question, how many cupcakes are left? Please type your answer into the box right here.
If you wrote 0, you're correct.
Zero is not a natural number. For example, when we think back to the kindergarten teacher example, she does not start counting her children starting with zero. She begins with one. You can think of the natural numbers as counting numbers, and when we count, we usually begin with the number one not zero. But zero does have some things in common with the natural numbers. We can talk about there being zero of some kind of object like cupcakes. When considered as a single group, zero and the natural numbers make up what we call the whole numbers. So let's go over the two groups we have so far. First, we talked about the natural numbers or the counting numbers, which are one, two, three, and so on, all the way up to infinity. So this includes a ton of other numbers, too, like 89, 560, and 9,345,760 as well as every other number that we could count from one up to infinity. The whole numbers include all of these numbers but also zero. So you can see that our sets of numbers are starting to build on each other. Which of the following sentences accurately describe the relationship between the natural and the whole numbers? Please select as many as you think are correct.
We know that the natural numbers start with one and increase by ones from there. So there are 1, 2, 3, 4, 5, and so on. Continuing to count up forever, and we know that the whole numbers are all of these numbers plus zero. So if we were to list all of the whole numbers in ascending numbers they would be 0, 1, 2, 3, 4, 5 etcetera. So that means that every natural number is a whole number, which is the second choice here and the last choice is also true because almost all of the whole numbers are natural numbers but zero is not a natural number.
So, you can see how this diagram that I drew earlier works. The circle representing the natural numbers is within the circle representing the whole numbers, showing that the natural numbers are all included in the whole numbers. If we pick any number, we can point to which category or categories it fits into by pointing somewhere on this diagram. For example, if I pick 18, then 18 is a natural number. So we can point here. But since that circle is inside of the whole number circle, we know that 18 is also a whole number. If we pick zero however, We can only point to the whole number circle, not the natural number circle since zero is whole, but not natural And if I pick a number like 16.5, well, that is not in either of these categories. So we would need to point outside of both of these circles. So you can see that we're building up this sort of world of numbers. And we're going to keep adding more to it. Next, we're going to talk about integers, which, if we represent them in our world of numbers here, would be drawn as a circle around the whole numbers like this. Even though I haven't told you anything about integers yet. Can you tell me just by looking at this diagram, which of the following statements are true?
Let's go through the same sort of thought process that we did for the previous quiz. Since the entire whole number circle is inside of the integer circle, we know that all whole numbers are integers. By the same token, some, but not all integers are whole numbers. We also then know that not all integers are whole numbers, so this answer can't be right. In this same way, not all integers are natural numbers since the natural numbers circle is inside the whole number circle. This also means though, that all natural numbers are integers. I hope this is helped you to see a bit more clearly how these different sets of numbers are related to one another and fit together in this diagram.
So now that we've talked about integers quite a bit, let's talk about which numbers actually are integers. We already know a bunch of examples of integers since we know that all whole numbers and therefore all natural numbers are integers. So any number that we could get to by starting at 0 and counting up by 1s is an integer. These numbers however are just the positive integers plus 0 of course. But there are also a ton of other numbers that are integers and these are the negative integers. The negative integers are just negative versions of all of the positive integers. So, since 3 is an integer, negative 3 is also an integer. The integers are all numbers from negative infinity, to positive infinity that can be written without any fractions or decimals.
Think about this new definition of an integer and about the other classifications of numbers that we've talked about. Please fill out the chart below. Please check off each of the categories the number falls into.
So now that you've completed your chart, let's take a look at the answers. Let's start out by looking at the numbers -1 and -9. They're both integer, and negative integer. Those are the 2 that should be checked. So now let's take a look at 0. There are 4 boxes you should have checked, for zero. You should have checked integer, zero, whole number, and non-negative integers. So now let's look at the number integer, positive integer, natural number, whole number, and non-negative integers for both of them. Let's look at the last category of numbers. The numbers 3/4, .5, and pi. Those don't fit into any category so you should have checked none of the above, for 3/4, .5, and pi.
We've been using the word set to describe these different groups of numbers and that's exactly what a set is, a collection of numbers. So, the natural numbers is the set 1,2,3 and so forth. And the whole numbers is the set 0, 1, 2, 3 and so forth. And we might represent the set of integers like this with ellipses on both ends to represent going on to infinity. We usually denote sets with these curly braces around the numbers and commas between the numbers we give certain sets. Like the ones we've talked about special names. But really a set can be made up of any numbers that we put together for whatever reason. So for example, this set -4, 7, 19 is a set. And so is -38, 56, numbers in the set of natural numbers are also in the set of whole numbers, we call the natural numbers a subset of the whole numbers. A subset of a set, is another set of numbers that are all contained in the original set. So its a little bit of practice to make sure you understand what sets and subsets are. Here's a random set of numbers, -4 -3, 0, 0.6, 1, 5, and 8. Which of the choices below are subsets of the set? You may pick more than one answer.
Now let's look that the answers. The first set, -4, 1, and 5 is indeed a subset of the original set because all of the numbers contained in the subset are in the original set. Let's look at the second subset, -3, -2, 0, and 5. Since -2 is not in the original set, this cannot be a subset of the original set. Looking at the subset zero. Zero is, indeed, in the original set. So that the set zero would be a subset of the original set. Let's look at the set. 0.5, 0.6, 1, and 2. There are a couple of numbers there that are not in the original set so that, that set would not be a subset of the original. Now let's look at the set 4,5. Again the number four is not in the original set so this is not a subset of the original. And the last set, 5 and 8. Both of these numbers are in the original set, so this is indeed a subset of the original set.
Since we're talking about sets and subsets, let's see how these words apply to the number groupings we've been talking about. Which of the statements below do you think are true?
To determine if these statements are true or false, it's somewhat easier to look at the diagram we previously discussed. So, if we look at the first statement, the integers are a subset of the whole numbers. We can see that this would be false because the integers is a larger set than the set of whole numbers. The whole numbers are a subset of the integers, that's true, as is the next statement, the natural numbers are a subset of the integers. However, the whole numbers are not a subset of the natural numbers. You can see the natural numbers are actually a subset of the whole numbers, which makes the last statement true as well. However, if we look at the statement, the integers are a subset of the natural numbers, this is also false. Because as you can see, the natural numbers are a subset of the integers. Overall, it's easiest to determine if a set is a subset of another set by looking at the circles. If the circle is entirely contained within another circle, it is a subset of the set.
So far we've talked about a lot of numbers, both positive and negative, and our world of numbers is looking pretty great. Still, we've really only been counting by ones, whether above or below Thinking back to your famous birthday party, what if your friend had baked a few cakes instead of making a ton of cupcakes? If you asked someone how much cake they wanted and they only knew about integers, they would have to say that they either wanted no cake at all, or they wanted 1 whole cake, or maybe even 2 entire cakes. This doesn't seem very practical and you'll also run out of cake very quickly. You would probably want to cut each cake into slices and then give each person just a piece of the cake. Let's say that you cut 1 of the cakes into 8 pieces. If you gave 5 friends each 1 of those slices, how much of that cake would be left over?
If we started out with a cake that has eight pieces and five friends, ate, a piece each, then there would be three pieces that were left out of the total eight. So, 3/8 is your correct solution.
Was the answer to the last question an integer?
No, 3/8 is a fraction that we could also write as the decimal .375, which lies between two integers, 0 and 1. This requires a new type of number classification. 3/8 is what we call a rational number.
We just said that 3/8 is a rational number, but let's talk about a more general definition for this new set of numbers. A rational number is any number that we can write in the form A over B, where both A and B are integers and B is not equal to 0. This is why these numbers are called rational numbers. Because they can be written as a ratio. Think about this for a second. Plug in several different combinations of integers for A and B to give yourself an idea of what some different rational numbers are. Then tell me which of the following world of numbers diagrams correctly shows the rational numbers?
This is a slightly tricky question, so if you found it a little bit difficult, no big deal. If we plug in 2 and 3 for a and b for example, we get 2 3rds or point 6 repeating as a rational number. And if we plug in 8 and 7, we get 8 integers. But, if we plug in 10 and 2 for a and b, we get 10 over 2 which is just equal to 5 which we know is an integer. So it turns out that all integers are rational numbers. And some numbers that are not integers are also rational numbers. And that means we need to draw the circle for the rational numbers, so that it completely surrounds the integer circle.
Now that you know that every integer is a rational number. That means we should be able to write every single integer in the form of a rational number. Which remember is a/b, where a and b are both integers, and b is not = 0. So how could you express -17 in this form? Please enter values for a and b to make a fraction equivalent to -17 in the slots below. There are a ton of different ways you can do this, so just pick one combination of a and b that you think will work and try it out.
Like I said in the question video, there are many, many different values for a and b that you could enter here. But perhaps, the simplest way is to let a= -17, and to let b=1. This is an easy way to express any integer. We can write 3 as 3/1 and 29 as come up with many other fractions that reduce to these integers if we multiply the numerator and the denominator, or a and b, each by the same number. In the case of -17, for example, if we multiply the top and bottom by 2, then we get -34/2, which is another acceptable answer to this quiz. Or you could have put 51/-3 or 17/-1. If you're feeling at all shaky on fractions right now, please take some time to do some of the practice problems we've provided on fractions or watch some of the supplemental material we've linked to.
Another question for you. Do you think that 0.6 is a rational number?
And the answer is yes.
Since 0.6 is rational we should be able to write it in a form of A over B. What is one way you could do that?
could've had this as your answer. 6/10 reduces to 3/5, so this is an acceptable answer as well. We can write many different decimals in fractional form, and thinking back to our definition of rational numbers, any decimal that we can write as a fraction is rational.
We saw in the last quiz how important it is to know what rational numbers look like when we write them with decimals instead of with fractions. How would you write If you can write the entire decimal in five or fewer digits, then enter the whole thing, and otherwise, round to five decimal places.
When we divide 7 by 16, we get 0.4375. And when we divide 1 by 3, we get 0.3 repeating. If we round this to five decimal places, we get 0.33333. Though, of course, this is just an approximation of the actual fraction. So with this problem, we see two different types of decimals. Terminating decimals like .4375 and repeating decimals like .3 repeating. Decimals of these 2 types can always be written as fractions. So that means they are rational numbers.
Now that we've added another set of numbers to our world of numbers, which categories do each of the numbers below fit into?
This problem probably looks familiar, we've just added a new category of rational numbers. In this case, all of the numbers are rational, with the exception of pi and the square root of 12. Neither can be expressed as a ratio of integers. We'll discuss these more in a few minutes.
A ton of the numbers that we encounter in daily life are rational numbers since we can count many things or just can split them into parts that we can measure, but there are certain situations that might not be quite this simple. Let's say that you have a little cousin and she wants you to build her a slide. This isn't just any slide though. Your cousin is very particular. She wants the slide to be exactly eight feet high, and she wants it to go from a trainer backyard, down into the lake by her house which is 12 feet away from the tree. Now you need to figure out how long the slide will be so that you can buy enough material to make it. You know that there is a rule for the right triangles, the Pythagorean Theorem, that says that the side lengths of a right triangle, a, b, and c, where c is the longest side, fit the formula a^2+b^2=c^2. Since the slide, the tree, and the ground make a right triangle, with the slide itself as the longest side, you know that you can begin to look for the length of the slide by first taking 8^2+12^2, which is just 8 times 8 + 12 times 12. For starters, what is this number? Please type the number of your answer into the box right here. Remember to do your multiplication before your addition.
If we first take 8 times 8 we get 64, and if we calculate 12 times 12 we get 144. If we then add these together, we get 64+144 are 208, great. We will make sure that we give you plenty of opportunities throughout the course to practice doing calculations like this just in case you found this tricky.
So thinking back to our slide triangle, we know that 208 is not actually the length of the slide that we're trying to find. Instead, this is the square of that length. In other words, we want to find out which number, when multiplied by itself, is equal to 208. Is this number rational?
And the answer here is no. The square root of 208 is 14.422205 and I could go on. I could keep writing, but this decimal actually extends on and on forever,without any sort of pattern that repeats. This is different from any of the other kinds of numbers we've talked about so far, and all of those numbers have been rational numbers. Now we need a different category, the rational numbers.
When we talked about rational numbers, we said that they are numbers that can be written as ratios of integers. Irrational numbers are numbers that are not rational, so they are numbers that can't be written as ratios of integers. We've seen that rational numbers are either integers or have decimals that either end or repeat. Irrational numbers, therefore, have infinitely long decimals that don't follow any sort of pattern. Perhaps the most common way that we run into irrational numbers is through taking square roots of numbers that are not perfect squares, just like we tried to do with the slide example. There are some other numbers that have special roles in math and physics that are also irrational, like the number E, which is about equal to that I have just rounded off these decimals because I don't want to write any more numbers, but like all the other irrational numbers, pi and e have infinitely many numbers in the decimals and they do not repeat. So now we've developed a whole world of numbers that fit most things that we need to count, measure, and calculate in every day life. There are still more types of numbers that we haven't touched on yet. Let's look at the whole entire world of numbers just for a second. Let's start with the natural numbers, such as 3 and 718. We then went from the natural numbers to the whole numbers by adding 0 to that set. And on to the set of integers by adding the negative whole numbers. From the integers we get to rational numbers by creating ratios of integers with a non-0 denominator. Those numbers which are not rational are known as irrational numbers, such as e and pi. They have infinitely long, non repeating decimals. Finally, we get to real numbers, which include the rational and irrational numbers we have discussed this far.
So for the next part of the course, we're going to talk about one of the ways that most people use math really frequently, and that is with money. Now even though these dollar bills came out of my pocket, we're not going to talk about my personal financial situation. Instead, we're going to talk about the financial situation of a pretend startup called Gleaming Glasses. Now this story may not be entirely fact based, but the same principles and the same math, that we'll talk about in regards to Gleaming Glasses, apply to businesses in the real world too. My friend Grant kept getting tired of having to clean his glasses. It was really tedious and frustrating. He'd have to take his glasses off of his face to clean them, which means that he could never really see how clean he was getting them in the first place. One day while in the car, Grant had an epiphany. If windshield wipers could clean his windshield so well, why not just make windshield wipers for glasses? But Grant had a problem, he didn't have very much money. So Grant decided that he would pitch his idea to a venture capital firm. Basically, a venture capitalist is a person who invests money in brand new companies that don't have enough money on their own to do what they want to do. Grant's glasses wipers idea was so awesome that he received $3,000,000 from a venture capital firm. Now, what matters isn't just how much money Grant got initially from his funders though. What he really cares about at any given time is how much money he has left over.
So, let's have a quiz. How much money do you think Grant has left at any given time? Please pick what you think is the best choice down here to replace the question mark right here.
So we know that grant started off with a certain amount of money from the venture capital funders. But then he doesn't just keep it, he starts to spend it. This means that the right choice must take both of these into account, the money that he was given initially and the money that he spent so far. The second and last choices only have one of these two things each, so that means that neither of them can be right. So I'm going to cross them both out. So that just leaves us with these two. Money from funders plus money spent, or money from funders minus money spent. Spending money, actually takes away from the amount of money that we have. It doesn't add to it. So that means that the third choice, right here, money from funders minus money spent must be the right answer. Now without even trying we've actually just done some math. I know that there aren't any numbers here, but what we've created is called an equation. It's called that because of this equal sign right here in the middle. Over doing with this equation is showing that these two quantities, the one on the left side of the equal sign and the one on the right side. Are equal to each other, or have the same numerical value. This is the first of many, many equations that we're going to look at throughout this course. And over time you'll learn to do a ton of different things with them.
So we have this lovely equation, which is a great start, but it's still a pretty general equation. Money spent over here, looks like just one lump sum. And chances are, that Grant's not going to spend all of his money at one time. In fact, it turns out that he's made a budget for himself, where he allows himself to spend $100,000 every month. So this means that over time, as each month goes by. The total amount of money that he spent will change. So this means that we can rewrite money spent over here, in a different way, using this new information. So, which of these choices do you think would be a new but accurate way to write out the amount of money that Grant has spent total at any given time?
So, we said earlier that Grant spends the same amount of money every month, which means that the total amount of money that he spent will increase incrementally as each month goes by. We know then at the total amount of money spent can't just be equal to the money spent each month Our answer down here has to involve time too. For example, after one year Grant will have spent way more money than he had after just one month of being in business. So that means that this first answer right here can't be right, because it doesn't involve time. However since we know that the same amount of money is spent each month, the second trace rate one here is the correct one. It says that the total amount of money spent after a certain amount of time is equal to the money spent each month times the number of months that have gone by. These 2 bottom answers don't involve the amount of money that Grant is spending, so we can just eliminate them also for good measure. Great job on this quiz.
So, here are the two equations that we've come up with so far. One equation for the amount of money that Grant has left after a given amount of time and another equation for the total amount of money he spent depending on how many months have passed since he's been in business. One thing that we can notice about these two equations is that money spent is involved in both of them. Now, remember that what an equation means is that whatever appears on one side of it is exactly equal to what appears on the other side. So since we know something about what money spent is equal to, anywhere we see money spent in any equation in the system, we can replace it with what we know it's equal to. Which is this, the number of money spent each month times the number of months. So taking this idea of substituting in values for other things that we know that they're equal to. Which answer down here would give us another way of writing out how much money we know that Grant has left? I know that this is a lot of writing to have on the screen at one time, so not a problem at all if this looks confusing. Just give the quiz a try and we'll see what happens.
So as I said earlier, what we're dealing with here is two equations this one and this one that have one common term which is money spent. We just said that we can substitute money spent with money spent each month times the number of months whenever we want to. So that means that this quantity down here On the right side of the second equation, can be moved up into this slot in the first equation. Having done that substitution now, we can see that the amount of money that Grant has left is equal to the money from the founders - the money spent each month the number of months. If we look down at our answer choices, that is The third choice, so great job on this quiz whether you got it right or you didn't. Like I said earlier, I know that this is a lot of writing to look at on the screen at one time, we're actually going to address that problem in just a second.
So now we have this awesome more detailed equation, but I still don't see any numbers. What do we know about numbers in this situation? Well we learned from our story that Grant earned $3 million in venture capital funding. So that means that money from funders right here is well equal to $3 million. So that means we can write down that money from funders is equal to $3 million. We also learned something else about numbers. We heard that every month, Grant spends a certain amount of money and in fact that money is equal to $100,000. Like we've been talking about for the past few minutes, any time you have two things on either side of an equal sign, you can substitute them from another in other equations. So let's see if you can figure out another way to write an equation for the amount of money that Grant has left. This time using these new numbers. So down here I'd like you to fill in which numbers belong in these two spaces that I have given you. Now you don't need to write dollar signs because I already filled them in for you. So we're just taking all of this information up at the top and combining it into one equation.
So, here we have another case of substitution. We know that money from funders is equal to $3 million. So wherever we see money from funders in the top equation, we can replace that with $3,000,000. So down here, we can write, 3 million in the first box, and we can do the same thing with money spent each month. We see that right here, in the first equation, and we know that this is equal to $100,000, so, we write $100,000 right there. So to summarize this, we can see that the money that Grant has left over at any given time is equal to the initial $3 millions that he got from venture capitalists minus the $100,000 he spends each month, times the number of months he's been in business.
So now we are in a really great position. We have an equation that actually has numbers in it. However, it's time for me to make a confession. I'm really, really lazy. I'm just super tired of writing out all these words in our equation. I know that it is important first to keep in mind what are equations about. But I think that we can remember that. Even if we do take a few shortcuts as long as I remember what belongs in this spot of the equation or what this part of the equation means. I might as well rewrite money Grant has left with something shorter. I've kind of already taken a few shortcuts already. For example, I've used a dollar sign instead of writing out the word money. And I used a pound sign instead of writing out the words, number of. So it's okay to take shortcuts, as long as they make sense to us. So the thing that I decide to replace these words with could be, really whatever we want. A single letter, or a single symbol. I think I'm going to use a, square. We can do a similar thing for number of months over here. Going with the geometry theme, I am going to call this, triangle, instead of number of months. So what I'm really doing here then, is assigning a new name, to each of these parts of the equation. We're going to say that square equals money Grant has left, and triangle equals number of months. So lets rewrite this top equation up here. Using our new symbols. Here are four spots that we need to fill in to make our new equation. or rather, our modified version of the original equation. Please pick from the buttons under each slot to select which number or symbol belongs in the box above. Good luck.
Now, the only thing that we wanted to change with this new equation was to plug in the symbols square and triangle for the words that we had before. So, that means that we only want to change what's right here and what's right here. That means that we can keep the numbers that are already in the equation in the same places. So let's just go ahead and bubble those in. Our substitution rule said that we can replace money Grant has left with a square and number of months with a triangle, so let's bubble those in. Money Grant has left comes first so that means that this square comes first And number of months comes very last. So that means, the triangle, comes at the end. So now, our equation reads, square = $3,000,000 - $100,000 times triangle.
Actually, as much as it might surprise you, we've been doing algebra right here. These 2 symbols, the square and the triangle that we assigned to replace the words that we had written earlier are called variables. In algebra we make a distinction between numbers and variables. Numbers are well, numbers, like 3,000,000 or 100,000, and variables are generally represented by symbols or single letters. A variable doesn't have a set value like a number does. Instead, it's value is allowed to vary depending on what we decide that it should equal. We'll talk more about how we make decisions about that soon. Numbers will appear either as constant terms on their own, just numbers like 3,000,000 right here, or in terms that combine numbers and variables like this last one on the right. When multiplied by a variable or set of variables, a number is called a coefficient. So again, going back to our variables, I could of chosen any two symbols I wanted to represent these two quantities right here, the square and the triangle. I didn't have to choose the triangle and I didn't have to choose the square. In fact picking these two particular symbols, is actually pretty unconventional in math. To give you a preview of what you're usually going to be seeing during this course, I'll write this equation out with more typical notation. We'll most often see the letters X and Y as your variables in equations in algebra. You can also see I've removed the dollar signs and the commas from the original equation down here to make this equation look a little bit more clean and strictly mathematical. You'll notice that I also sneakily deleted the multiplication sign we had in the top equation, and didn't write it down here at the bottom. When we have single terms that involve both numbers and variables multiplied together, we don't usually write the multiplication sign in, like I did at the top. As long as we write the symbols directly next to each other, with nothing in between, multiplication is assumed to be happening between them. It's like there's an invisible multiplication sign written right here. I'm going to erase that though, to remind us that we're not usually going to see any multiplication signs written between numbers and variables. Although people doing math often use the variables y and x in their equations, I really want you to remember that this is not necessary. Triangles and squares are just as useful as x's and y's. No matter what symbol you choose to write a variable with, the power that it has is still the same. It acts as a slot where we can insert different numbers to make terms and expressions take on different values. So maybe I set this first variable equal to 3 and the second equal to -1 or maybe this is -1/2 and this is 23. In certain situations, there will be restrictions placed on what different variables can equal. But, until we have more information about what they equal, we leave them written as variables, showing that their values are still sort of a mystery to us.
To make sure that you know how to recognize variables, let's have a quiz. Let's see if you can identify which things in these equations are variable terms. So please place a check under every single term down here that has a variable in it. Remember for our purposes right now a variable is anything that is not just a number.
Starting with this first equations up here, we see that we have 3x+4 equals smiley face. 3x has a 3 in it, but this is multiplied by x. A term like this, even though there is a number in it, is called a variable term. Because its whole value is going to change or vary depending on what number we plug in for x. This is definitely a variable term, 4 is just a number so no check there and finally, we have a smiley space. That means, it must be a variable. And we can go through the other three equations in the same way turn-by-turn to figure out what should be checked. And know that there are some things down here that we haven't talked explicitly about yet, and that might have thrown you off. Like, we have a fraction right here and we have a couple of expononents, but we'll get to those things pretty soon. And understanding the difference between variable and nonvariable terms is really important for understanding everything that we're going to do in Algebra. That means we're off to a great start in developing our foundation.
In the past few minutes we've touched on a lot of concepts that are going to play a really central role in the rest of the course and really in all math going forwards from here. So I would like to do a review of the algebra vocabulary that we've covered so far. I think it's actually going to amaze you how much we've already gone over in a pretty short amount of time. So first thing first, we have numbers, as one might expect in math. So what are some examples of numbers? Well, we could have any number of things. So like we learned in the very beginning of this course, they're all different kinds of numbers. Some that are pretty simple and we use all the time, like 0 or 1 or special, like the root of 2 or pi. Then in contrast to numbers, we also learned about variables. So, here we have all different kinds of symbols, we have some letters, we have some sillier things like hearts and smiley faces, we got some greek letters. But regardless of what we write variables as they all have the same purpose, as we learned earlier. Variables are symbols that we use in equations and expressions that serve as slots into which we can plug other numbers to make our expressions take on different values. So comparing numbers and variables, numbers have fixed values. They inherently show certain quantities. Variables, however, don't show outright what they stand for, they're just place holders into which we can insert different numbers. One thing to take note of when we're looking at what's inside each of these circles is that we actually have a symbol on the number side. That's this pi right here. Since pi is an irrational number, it extends for an infinite number of decimal points. Now, it took me an awfully long time to write all of this out. So if I were to write even more, that would take me even longer. To make our lives much more simple, we give this infinitely long number a special name, which is pi. A similar thing happens when we write square roots. We have a number, root sign over it. Since the root of 2 is also an irrational number the square root sign is just another way of taking a shortcut to not have to write out an infinite number of decimals. So bottom line, when you see symbols, they're almost always variables. They'll be certain cases. Which you'll become well acquainted with when you see things like pi or like a square root sign that are symbols of some sort but represent numbers.
Now as I mentioned earlier variables and numbers can be combined. And when we do this, we create expressions. An expression is what you get any time you put numbers and variables together, with mathematical operators. So those are things like plus minus divide and multiply. To make things a little bit more concrete I'll write out a few examples of expressions. So in each of these expressions right here, you see some terms that have variables in them. Like 3x or, or -4m^4 / 13n^3. And these are combined with terms that are just numbers, like 4 or -2. The separate things that we add or subtract to come up with the expressions are called terms. Any term that has a variable in it is called a variable term like 7.8y / 3 or z and other terms that don't have variables in them that are just numbers are called constants or constant terms. I've only labeled the types of terms in the first expression but we could easily figure out what kinds of terms are in the other two expressions, as well.
Now, we can't forget about Grant's gleaming glasses. So, let's see how the vocabulary we've discussed pops up in the math we've done for him so far. Now, let's see, what was that last equation that we wrote for the amount of money that he has left? That's right. We had y = 3,000,000 - amount of money that Grant has left and x over here stands for the number of months that have gone by since Grant started his business. So you have 3 terms in this equation. We have y, 3,000,000, and -100,000x. Which of these terms are variable terms and which of them are constant terms? Please fill in the proper bubbles below each term.
Lets go through the terms, 1 by 1, to figure out what type each of them is. So first we have y, y is not just a number, so that means it cant be a constant term. And we also know, that y is a frequently used letter, to represent variables in Algebra. So, y is a variable term. Then we have 3 million. There are no symbols, other than numbers in this term. And we know that 3 million is a number, we know how big it is already, so this has a constant value, that means that its a constant term. And lastly we have negative -100,000x. Note that the negative sign belongs with the term itself, so we dont say that the term is 100,000x, we say that it is -100,000x, anyway, this term has two parts, it has a part with a number and a part with a variable. As we talked about before though if a term has a variable in it at all, then that term counts as a variable term. When we plug in numbers for x -100,000x is going to change values because of what we plug in. So the whole term despite have a number in it counts as a variable term.
Let's do one more short quiz with this equation. What would be the proper name for this number -100,000 that's part of this last variable term? So here are four choices. Is -100,000 in the context of this equation a variable, a constant, a coefficient or a term? Please select one or more of the choices below that you think are right.
This is actually a little bit tricky. Even though we've talked a lot about what vocabulary is, sometimes it's hard to apply it to actual situations. Let's start from the bottom and work up. -100000 is part of a term, but the term that it's part of is actually -100,000x. Remember that the term is the whole thing that's going to be added to other things or subtracted from other things. So -100000 on its own is not a term, so let's cross that out. We said earlier that -100,000x is a variable term as a whole, but it's not the -100000 that makes the term a variable term. It's the x that does that, so, -100,000 itself is not a variable. Constant was the word that we used for numbers that were terms on their own, things like 3 million right here. This is something that is a term by itself that has no variables as part of it. -100,000 is not standing on its own, it's multiplied by x, so this is not a constant either. That leaves us with coefficient. Remember that coefficients are the number parts of variable terms. So they are numbers that are multiplied by variables to create variable terms. I know there are a lot of subtle differences involved in this, but over time this will become a much more natural feeling. Great job working on all this vocab.