this video is about section p one real numbers and their properties so we'll first start off with the definition of a set and then we'll see a bunch of different sets so a set is a collection of objects and the objects that are inside of the set those are called elements examples of sets include things like the set of all things inside of your backpack the set of all students in a class the set of all phone numbers you have in your contact list so in this class we're going to be dealing with some important sets that involve numbers so let's start with number one the set of natural numbers this is kind of long to write so you can abbreviate this by using the capital letter n and what's in here so you can list numbers that are in here there's one it's the smallest thing that's in here two three four five six etc so the natural numbers are the numbers that you count with when you're counting dollar bills the second important set we'll deal with is the set of whole numbers and we will denote this by the letter capital w and what is in this set you take all of the natural numbers throw them in there and then also throw in the number zero so you have 0 1 2 3 4 5 etc and now the set of integers which we will denote by the letter capital z to form the set you take all of the whole numbers throw them in there and then also throw in all of their negatives and the ellipses on both ends mean that the numbers go on to the left and to the right forever the next set that we want to talk about is the set of rational numbers this is denoted by uh capital q and there are so many of these that we cannot list them like in the previous three sets so instead here we're going to write a description of what it means to fall into the set and this is any number that can be expressed as a as a ratio of an integer divided by a non-zero integer so here's the key word ratio so it can be expressed as an integer divided by a non-zero integer so for example seven over eight actually let's make it 18. 7 over 18 is a rational number because 7 comes from this list above and so does 18 and that 18 is not zero so this is a rational number and you can write that this is a natural number in shorthand by using this symbol and saying that it lives inside the set q so what else is an example of a rational number negative 5 is a rational number because you can write negative 5 as negative five divided by one now you could have also written this five divided by negative one but the point is that you can always express negative five as a ratio of an integer with a non-zero integer so negative five is also a rational number so what is another example of a rational number you'd have .33333 repeating and you know that this is a rational number because this is the same thing as a third and so point three repeating is also a rational number zero is a rational number because you can write zero zero over one um 2.125 is also a rational number because you can actually write this as 17 over 8. so you may notice that we have covered one two three four sets and so a natural question that you might want to ask yourself is how are these sets related and so it turns out that the natural numbers are all automatically whole numbers which are all automatically integers which are all automatically rational numbers and so you can write that using this sum this symbol so the natural numbers is a subset of the whole numbers the whole numbers is a subset of the integers which in turn is a subset of the rationals so now let's look at a set that is completely disjoint from these four sets and that's the set of irrational numbers and we are going to denote this by capital i so typically these numbers are they look like they are non that means that the decimals don't end non-repeating decimals that's typically what they look like in fact any real number that is not rational is automatically irrational and so some examples of this include pi so this is 3.14159 blah blah blah and these decimals go on forever so they do not terminate and they also do not repeat unlike 0.33333 etc or the negative square root of 2 this is negative 1.414213 five six two and this goes on forever so these are some examples of irrational numbers so the biggest set that we're going to cover in this class is a set that contains all of the sets above and that's the set of real numbers which is denoted by a capital r and this is the set of all rational and irrational numbers now you can visualize all of the sets above and the relationships of these sets by drawing yourself a diagram so we started off with natural numbers that's a set but all natural numbers are whole numbers so this set is contained inside of the set of whole numbers in fact the only difference between the two sets is the number zero zero is in the bigger set not in the smaller set now these two sets are contained inside the set of integers which in turn is contained in the set of rationals oops and then we also learned about the set of irrationals but the irrationals are all of the sets or all of the numbers that are not rational so you can draw that box over here this is the set of all the irrationals and then all together you take everything inside of this whole box and that set is the set of all real numbers so now that we know the world that we're operating in we're operating inside of the screen box we're going to discuss the properties that occur in there so these are the properties of r properties of real numbers so if you take any three real numbers um so all this means is a b and c are real numbers i just use the um this shorthand symbol so number one you get closure all this means is that if you take any two numbers in that box and you add them together you stay in that box in that green box also if you multiply any two numbers inside of this green box you stay in the green box so you have closure under addition and multiplication property two we have commutativity so the commutative property says that you can add numbers in any order property three we have the associative property it says that if you have three numbers that you are adding together you can associate them or group them differently but they have to stay in the same position so you can you can add b plus c first and then add a or you can add the first two numbers first and then add a c and the same holds true for multiplication a times the quantity bc is the same thing as multiplying a b together first and then multiplying by c number four we have the distributive property of multiplication over addition and this says that if you have a times the quantity b plus c you can distribute the a and get a times b plus a times c number five we have the identity property zero plus a is always going to equal a no matter what a is and this zero has a special name that is called the additive identity we also have the property that if you take one times a you get a and this one is called the multiplicative identity property six the multiplicative liquidative property of zero this just says that if you take any real number and you multiply it by zero you always get zero no matter what real number a is property seven every real number a has a unique additive inverse negative a in fact this negative a is also going to live inside of the set of real numbers and it has the property that if you were to take a and add it to its additive inverse you get zero so again this is called an additive inverse property eight every non-zero real number a has a unique multiplicative inverse and this multiplicative inverse is going to look like 1 over a and it's also going to live inside the set of real numbers and it has the property that if you were to take a and multiply by 1 over a you get 1. so this is called a multiplicative inverse you might be more familiar with the colloquial term which is the reciprocal so you can say multiplicative inverse or you can say reciprocal so we can do a really quick example here if you wanted to find let's pick a number so say you were to pick the number 13 if you wanted to find the additive inverse and the multiplicative inverse what would these be so the additive inverse would be negative 13. this is the number that you would have to add to 13 to get 0 and the multiplicative inverse would be 1 over 13 this is the number that you'd have to multiply to 13 in order to get 1. and notice that in property 8 we specifically say non-zero because otherwise you would not be able to take the reciprocal so the denominator is not allowed to be zero here so every non-zero real number has a multiplicative inverse but zero does not have a multiplicative inverse okay so now that we are completely familiar with a set of real numbers and we know about all these properties you can also compare numbers you can compare real numbers and you can stick them on a number line so the real number line looks like a line with a bunch of numbers on it and so the bigger the number is the more to the right you are so one is bigger than zero therefore one is to the right of zero two is bigger than one therefore two is to the right of one etcetera where would negative one be negative one is less than zero so it'd be somewhere over here negative two is smaller than negative one therefore negative two would be over here negative 3 is even smaller so be over here and so you can stick a bunch of numbers on the real number line let's identify a few more points on here so where would uh negative one half be negative one half is smaller than zero so it should be located somewhere over here negative one-half is bigger than negative one and so it should be somewhere in the in between negative one and zero so there's negative one half where is the square root of two the square root of two is around 1.41 so it is somewhere over here so now that you can visualize numbers on a real number line we can look at what's called the absolute value so the absolute value of some real number a is the distance from the number zero to the number a along the real number line so for example here's the real number line with zero let's put three on there what is the absolute value of three well according to our definition we'd have to measure the distance from zero to three and you can measure that in three units so the absolute value of three is three so what about the absolute value of negative three so negative three would be located over here to the left of zero and the absolute value of negative three measures the distance from negative three to and again that is 3 units so you can write the definition of the absolute value function more formally by saying that the absolute value of a is equal to one of two things if the number in between the absolute values is already not negative then the absolute value of that number is just that same number but if the number in between the absolute values is negative then you will throw a negative sign in front of that negative number to get the distance of a to zero which we'll make of course so this guy here is actually a positive number because we have negated a negative number so as an example of this you can look for the absolute value of negative eight and the way that you do this is you look at the number in between the absolute values that number turns out to be negative and because that number turns out to be negative we have to follow this rule which means that we put a negative sign in front of the negative eight so we negate what is in between the absolute values and we get eight as another example we could find the absolute value of 12. so you first look at what's in between the absolute values that number is already positive so that means that we're going to spit out the exact same number so this is equal to 12. so what if you want to find the distance between two points any two points on the real number line so the distance between two points x and y on the real number line is so the distance between x and y is going to be the absolute value of the difference between x and y and you really can subtract in this way or you can subtract in the other way it doesn't matter which one you do so for example say you want to find the distance between negative five and two so you want the absolute value of the difference between negative five and two so you can subtract this way or you can subtract the other way minus two minus five you will get the same answer so if we do it this way this is the absolute value of negative seven which turns out to be seven and you can double check that if you were to draw this so two would be to the right of zero and negative five would be to the left of 0. and so if you wanted to verify that the distance was indeed 7 you can check that to get from negative 5 to 0 1 two three four five you'd have to make five jumps and then to get from zero to two you'd have to make two jumps so the distance here is seven which means that this is completely consistent with our picture so the next thing i want to talk about are exponents these are also called powers um let's first deal with when the power is a natural number so we're going to pick a natural number which we'll call n and some real number which we'll call a so what happens if you take the real number and you raise it to that natural number so this is defined to be a times a times a and you keep multiplying a to itself until you have n instances of a so the exponent really tells you how many times the a appears as a factor so this a here that is called the base again the n is called the exponent or the power and the base is always the number that is sitting directly to the lower left of the exponent so let's do some examples negative 3 squared let's identify the base first and then the exponent and then let's do the calculation so what is the base the base is negative three the exponent is two and so if you take negative three and you raise it to the power of 2 you should get negative 3 times negative 3. so negative 3 is multiplied to itself it appears as a factor twice if you multiply these two numbers you get 9. now compare this to what happens if the parentheses aren't there so negative 3 squared what is the base what is the exponent so the exponent is the same it's still two but the base remember is the number that is directly to the lower left of the exponent which means that the base is just three not negative three like in the previous example so that means that 3 is the only thing you'll be multiplying to itself twice but that negative sign still has to be there so now multiply negative 3 and 3 and you get negative 9. and you should notice here that our answers are completely different from each other you get 9 in the former and negative 9 in the ladder so as a general note it is important to note that 3 oh negative 3 with parentheses squared is not the same thing as negative 3 squared without the parentheses so now we know that we can add real numbers multiply real numbers subtract real numbers divide real numbers we can raise them to powers and so what happens if you were to mix up all of these operations and do them all in one problem which one would you do first so we have an understood list of priorities of what to do first this list is called the order of operations this is universally agreed upon in the world and by computers and calculators because we made them respect these orders so the first thing that you do if you have a bunch of symbols mixed together you first do what's in grouping symbols first do what's in grouping symbols so this is the top priority what do grouping symbols look like um grouping symbols could be parentheses brackets it could be curly braces they could be even absolute values so when you see anything like this you do that first and if they're nested so that means you have parentheses inside of a bracket inside of another set of brackets or parentheses within parentheses within other parentheses then you do the innermost group first and then work your way out number two if you don't have any grouping symbols or you're already finished working with them the next thing you should scan for are exponents and you do those the third priority after grouping and exponents after those are done you then scan for multiplication or division and the these are done left to right so if you're reading from the left and you see multiplication first you do multiplication first if you're reading from the left and you see the division symbol first then you divide first number four the least priority is addition or subtraction how do you know which of these two to do first you do these left to right so if you see addition first when you read from left to right you do addition first and if you read subtraction first from the left then you do the subtraction first so let's do an example 15 divided by 2 plus 3 minus 2 times 5 plus 3 squared [Music] first thing we do is what's inside of the parentheses so this is 15 divided by 5 and then everything else where you don't do the calculation just copy it down and then this is equal to so what do we do next we do exponentiation so we have an exponent there three squared at the end so we'll do that next again everything else you leave and then the next order is division so we divide 15 by five that gives us three minus two times 5 plus 9 and then if you scan that line it looks like you have a multiplication to do so multiply the two and the five together even though it's very tempting to subtract the two from the three don't fall into that trap so do the multiplication first and now you have subtraction and addition together do the subtraction first because it is uh the first operation coming in from the left so this gives you gives you negative seven plus nine and the sum of these two numbers is 2. now knowing the order of operations is especially useful when you're trying to plug in into an algebraic expression so let's do an example of that so let's say that you know that the value of a is negative one and the value of b is negative three and the value of c is four let's try to find b squared minus 4 a c so that means you're going to replace every variable with the corresponding value so the b has to be squared that means that the negative 3 has to be squared it is very very important that you put the parentheses there minus 4 times a a is negative 1 times c c is 4. and now let's keep simplifying this negative 3 squared that is the first thing you do because you have an exponent there so this will give you nine because negative three times negative three is nine and now if you look at what you have you have you're subtracting four you're multiplying by negative one and then you're also multiplying by four so what do you do first remember that multiplication takes precedence over subtraction so you're going to first multiply the four and the negative one and that will give you negative four times four you'll get 9 minus now multiply let me use a different color multiply the negative 4 and the 4 and that will give you a negative 16. now add these two things together and you get 25. and finally the last thing that's covered in this section is simplifying algebraic expressions and the idea is pretty straightforward you can add or subtract like terms that is you can add or subtract terms that have the same variables and the same exponents on those variables so it's very very important they have to have the exact same variables so whatever variables occur in one term they have to appear in in the other term and not only that but all of those variables have to have the exact same exponents and when this happens you can combine them by adding or subtracting so let's do a quick example number one 17a plus 16a 17a and 6a are like terms because they both have a variable of a and the exponents on both terms so there's an invisible a to the there's an invisible one there on the exponent so the variables a are the same and the exponents on that a is the same so by the distributive property this is the same thing as 17 plus 6 times a this is by undistributing the a you can add the 17 and 6 and you'll get 23 times a number two for a b plus 7 b minus 2 a b so if you look at these three terms let me write this guy again if you look at these three terms um which ones are like terms well only the first and the third are like terms because the middle term doesn't have an a like the other two so you can change the order of the minus 2b and 7b and then you can combine the first two terms by noticing that they both have an a b and so you just have to subtract the two from the four plus the 7b at the end so this will give you 2 a b plus 7b and you know that you can't go any further because 2a b and 7 b are not like terms and they're not like terms because the first term has an a and the second term does not have an a and finally example 3 negative 4 times x plus 2 minus parentheses 3. let me write that again 3 minus 5 x how do you simplify this so the first thing we should do is use the distributive property so we're going to distribute the negative four so negative four x minus four times two again that is by distributing the negative four minus there's really an invisible 1 here so you can distribute this negative 1 multiply that in so this is going to be minus 1 times 3 minus 1 times negative 5 x and keep going this is negative four x minus eight minus three plus so y plus so here i'm multiplying the negative 1 times the negative 5x so plus 5x and now group together the like terms the like terms here are let's see negative 4x and 5x plus 5x these are like terms because they both have an x on the other hand the negative 8 and the negative 3 are like terms because they do not have x's so combine the negative 4x and the 5x and then combine the negative 8 and the negative 3 and you can simplify this further you will get let's see negative 4 plus 5 times x that's by the distributive property minus 11 and negative 4 plus 5 is 1 but when the coefficient of x is one you don't need to write it so you can actually omit this one you can make it invisible and so your final answer for this problem is x minus 11.