hey guys welcome back this is Jonathan Gardner lecturing on circulates basic mathematics the first section of the first chapter is called the integers and like most math textbooks or reintroductions or remedial math level to get you back up to the speed of calculus we need to first talk about what numbers really are so the numbers that you're familiar with that you learned ever since you were little like 1 2 3 4 etc these numbers they're all called the positive integers let's see if you can see that color as you can tell that the series goes on forever it doesn't end it's an infinite series at some point in your education you learned about the special number called zero zero is special because it represents nothing and I don't know if you've seen videos on math about the the breakthrough that zero was but this is a very important and significant number it's going to show up again and again as we learn math it has some very special properties all right now if to understand these numbers this his example in the book is hilarious he says these numbers could represent the score you get on a test in math you know if you got a hundred that means you got all the answers correct if you got one that means you only got one of the answers correct but if you got a zero that means you got nothing right and you don't know anything about math and but hey we're here to learn right so we can represent these numbers as if they're on a line let me use a fuchsia or pink or right I don't know what color this is so we draw a line and then on this line we choose some arbitrary point to be our zero I'm going to choose this point to be zero and then we stretch out a unit distance just it doesn't matter how long it is and we call that one and then two of those distances we call two and three of those distances we're going to call three and this number line is an important concept it's a really great way to visualize and understand what the numbers are let's use this peach color here that it's called the number line but one thing I want you to think about is that the number line isn't really representing positions in space aside from where the zero is the zero tells you that on this spot on this piece of paper this number line begins these numbers tell you a distance from that position so three really says you've gone three units to the right and you can see there's one two three units in between here and three and two says you've gone two units to the right there's zero and the positive numbers together we call these the natural numbers why we call them that because they arise naturally a lot don't get too concerned about the names we choose for things it's not terribly important we just give them names to make sure that they're different and we know what we're talking about when we talk to each other this special point here where the zero is we call that the origin we'll be talking about the origin a lot all throughout math and the natural numbers can be used to measure other things he says in the book for example a thermometer you can use the natural numbers to represent that how hot it is outside if you're in Celsius land the zero degrees would be freezing and a hundred degrees would be the earth is boiling but if you're in fahrenheit lines zero degrees would be below freezing well below the freezing and 100 degrees would be a very warm day a hot day so that means that these numbers it depends on what you're talking about and how they work well on the thermometer if it's a very cold day in Celsius land or in Fahrenheit land then you notice that the durometer drops you can go down to negative one you go down to negative two and what does that mean the negative numbers means you're moving that many units to the left I kind of made that one a little shorter or smaller it should be the same size as the other ones and this obviously goes off in to negative infinity and so these numbers negative one negative two negative three etc we call these the negative integers let me write that down for you these are the negative integers let me use the pink again or that peach negative ha so now you know about negative numbers that's basically everything you need to know about them we're going to learn what you can do with them later where this this conversation that we're having right now is something that we're gonna have again again in math where we introduce a new type of number something that we can do math with we talk about where it starts how it moves how you can go from one number to the other and then we begin a discussion about the operators we can apply to the numbers in the case of integers there's a very important operation you can use it's called addition I'll write that in black here because it's important and you learned about addition at a very young age if you had two pieces of candy and your friend had two more then if you took his candy you'd have even more than two you'd have four right and so we write down addition like this so we take some integer we put that little cross sign and then we put some other integer in this case seven and then we have these two bars here and then we write 12 and let's analyze what this sentence says and this these equations are just sentences and if you go back to the old math papers you'll see that people actually spelled out in their native language the math sentences they didn't use symbols this this symbol right here actually comes from a Latin word that means and right so this means add this means equals okay so adding says that you're taking this thing on the left and you're taking that thing on the right and you're bringing them together like you would pieces of candy the equals says that both sides are the same they're balanced right so if there's 12 things on this side that means there's 12 things on that side and you can switch the equal sign anytime you want we call this reflexive property things that are equal can be flipped around like that and it doesn't change anything so 5 plus 7 equals 12 and you learned that in elementary school but let's learn some more interesting things about addition the first interesting thing we're going to learn about addition is that when you take zero and you add any number and how do you write any number well in this case I'm going to write the letter A okay and what this a means is it could be any number in there and then we're gonna write this equals what if you take zero and add it to any number Y you're going to get that same number back okay so this means any number and this means the same number this is a very important property of addition and we'll be using this all the time and we can even write it this way too we can say you start with any number and you add zero it's the same as adding zero to any number either way and it's the same as adding or just having that number by itself I'm gonna draw a box around this okay and in the book he labels this equation in one okay in math textbooks we label our equations and we refer back to them constantly so you have to remember which equation is which and if you need a flash card or if you just need to be able to jump back in the book and remember what they're talking about when they say in one then you need to do that you so we're going to talk about N 1 the equation N 1 or this is the equation we're meaning that when you add 0 to any number it doesn't change it okay now remember the equal sign works in Reverse so if you're starting with a then you can put a 0 on the left side of that or a 0 on that right side and add it together and you won't change the value of a it won't change how many numbers you have the next question that you're probably having is what happens when you add negative numbers what does it mean to start at let's say 10 and add negative 5 what does that do what does that mean write the parentheses here don't get confused the parentheses just means we're gonna do what's ever on the inside first and then we're gonna do what's on the outside later okay so this means that we're gonna this is just saying that minus sign there is saying that this 5 is not a positive I was a negative 5 but this isn't the minus sign you're used to but let's let's talk about the - I'm really quick so if we have ten minus five what is that equal well that means we're starting at ten and we're taking away five things well how many would you have left you'd have five left right because and the proof of this is that five plus five equals ten right we know that five plus five equals ten so if we take away five from ten then we have five left over so this sentence is related to that sentence in it in a way that we're going to explore maybe in the next video I don't know okay so what happens if we did this though if we started at ten and we subtracted 15 what would we get that's an interesting question let's draw the number line over here we're gonna draw it with a new origin where to put the origin and say here so we have one two three four five six seven eight nine ten and the ten here means that you go ten to the right this number line goes on for infinity of course so we're at ten to the right what does it mean to subtract five well if you subtract five you go back five units so you have 10 nine eight seven six five you can see there's one space 2 space 3 space 4 space five spaces so subtracting or adding the negative means you're moving to the left adding means you're moving to the right okay so let's move to the left 15 units so we move 10 units we're back at the origin 11 12 13 14 15 well it's minus 1 minus 2 minus 3 minus 4 that's minus 5 so the answer 10 minus 15 is negative 5 okay that doesn't sound very complicated does it let's do a couple others as examples so if we had 7 and we added negative 3 then we would get what so we start at 7 5 6 7 and go back 3 1 2 3 we didn't a bit before right let's do another one these are in the book I'm just duplicating what's in the book so I don't ruin your study you should go back and review the books what if we start at 3 so 1 2 3 and we go back 5 1 2 3 gets us back to 0 4 or 5 takes us to negative 2 okay here's another interesting property let's go to a new page to talk about this property if we start at five slide that down so you can see and then we subtract five that's the same as starting at five and adding a negative five and what does that give us it gives us zero what if we started at negative three and we went three to the right where does that get us zero this tells us that when we start with any number and we add its negative counterpart it's the same as starting at the negative counterpart and adding that same number it's the same as zero this is equation in two and this is a very important property that we use all the time when we do math so just just like in in one we discovered that we can add 0 to any number and n 2 we discovered that we can get to zero from any number positive or negative by adding the inverse then the additive inverse what we can call it on the number line what does this look like pull out my little pink we have the number line here goes off to the right goes off to the left let's put 0 in the middle this time if we start at a and we go back negative a steps we get back to zero and if we start at negative a and we go forward a steps we get back to zero this is sort of a mirror with the origin being the plane of the mirror so the reflection when a looks in the mirror it sees negative 8 and when negative a looks in the mirror it sees a so the positive and the negative realms are sort of mirror images of each other one goes to the right while one goes to the left all right note and this is important we say - a and we say negative five okay what is the difference well the difference is what if a was if a was let's say negative three then - a would be three minus a is not negative it's positive so if you don't know that a is positive you you can't say that this number is negative you have to say it's minus a it's important distinction you'll often hear people when they're doing their math lectures they'll say negative a when it's not really negative it could be positive and one more thing that we call that the additive inverse that's right so we'll say three is the additive inverse of negative three and negative three is the additive inverse of three that's the number a word that you'll hear again and again inverses additive multiplicative inverses and we'll get into the rules for addition in the next section thank you for watching take care of the line hey guys thanks for watching this video this video is part of my series on Surrey lanes basic mathematics you can click here to watch the rest of the videos in the playlist you can click here to learn more about me and you can click here to support my channel thank you so much [Music]