what's going on bessies today we're going to be diving back into the math portion of the exam for aits and we're going to be talking about arithmetic with rational numbers let's get started so let's start with order of operations we all understand how to perform addition subtraction multiplication division and so on now it's time to explore scenarios where these operations are going to be combined and applied simultaneously to establish protocol for executing multiple mathematical operations we use the proper sequence known as pemos This Acronym represents parentheses exponents multiplication division addition and subtraction guiding us through the sequence in which these operations should be carried out moving from left to right as we work our way through initially we are going to address any calculations within parentheses next we're going to handle exponents and then after this we're going to look at multiplication and division now it's really important to understand that we are going to perform this by moving from left to right neither multiplication nor division takes precedence over the other so if our first operation is division we would perform that operation first finally we conclude with addition and subtraction following those exact same rules neither addition nor subtraction are going to take precedence over the other so if our first operation is subtraction on the left we are going to perform that operation first let's let's take a look at an example of what this will look like on your t's so our first equation is 4 plus our parentheses 3 * 2 - 8 / 2 so the very first thing we want to do following PM dots is perform the operations within the parentheses we do this by multiplying 3 * 2 which is going to give us 6 next we're going to move on to exponents we don't have any so we can automatically skip this step then we're going to perform multiplic and division remember we're moving from left to right so the only multiplication and division that I can see is the 8 / two we're going to perform that operation and it's going to give us four and then lastly we're going to finish up with our addition and subtraction again moving left to right nothing takes presidence over anything else so in this particular case we have 4 + 6 that's going to give us 10 we're going to drop down our minus 4 and that is going to give us 10 - 4 is equal to six so the correct answer for this equation is going to be six let's take a look at another example before we move on to our practice questions so we have 15 + parentheses 3 + 2 raised to^ 2 - 9 * 6 + 2 ra to^ 3 so it looks a little complicated hang in there with me we're going to get through this so first we're going to perform the operations with the parentheses first so 3 3 + 2 is the only operations we see with inside our parentheses and that is going to give us five moving down next we're going to calculate our exponents that's our powers and our square roots so we have two exponents here we have 5 ra to the^ 2 and 2 raed to the^ 3 5 raised the^ of 2 is going to give us 25 and 2 ra to the^ 3 is going to give us eight now we're going to perform our multiplication and division moving from left to right so as we can see here we only have one multiplication and that is 9 * 6 so 9 * 6 is going to give us 54 and that'll end that portion of pendos lastly we are going to add and subtract again moving left to right starting from the left we have 15 + 25 that's going to give us 40 40 minus 54 is going to give us -14 and then we're going to add 8 and that is going to give us our correct answer of -6 let's start with our first practice question calculate the following operations we have 5 * parentheses 2 + 3 - 4 ra^ 2 + 6 so we're going to start out with our pendos with our parentheses so in parentheses we have 2 + 3 that's going to give us five so our new equation is going to be 5 * 5 - 4 ra the^ of 2 + 6 next we're going to move down to our exponents so we only have one exponent here right that's 4 ra to the^ of 2 so our new equation is going to be 5 * 5 - 16 that's our equation we just did plus 6 now we're going to move on to multiplication and division again moving from left to right so if we take a look here we only have one multiplication that's 5 * 5 so that's going to give us 25 - 16 + 6 and in our last step we are going to do addition and subtraction again moving left to right so we have 5 - 16 is going to give us 9 so we're going to drop down our our plus six that we have left over up here and 9 + 6 is going to give us 15 so the correct answer for this equation is 15 and yes there it is our correct answer is C next let's delve into the concept of rational and irrational numbers so at its core a rational number is simply any number that can be expressed as a ratio of two integers in the denominator cannot be zero right it's right in the name ratio rational it's going to be really hard for you to get forget that when you're taking your te's let's take for instance the number one it can be depicted as 1 over 1 or -2 over -2 or even 500 over 500 these these examples all demonstrate various ways to represent the number one as a ratio of two integers and theoretically there's an infinite amount of ways that you can do so so long as the numerator and the denominator are the same similarly the number -7 can also be expressed as -7 over 1 7 over -1 or even 28 over 4 the list of representations when it comes to -7 is also endless it's clear that -7 is a rational number because it fits the definition of being expressed as a ratio of two integers so what about numbers that aren't whole integers consider for example the number 3.75 how can we express this as a ratio of two integers one way we can do this is to look at 3.75 and convert it as a fraction that would give us 375 over 100 which is the same thing as 750 over 200 alternatively we can understand that 3.75 is 3 and 34s which converted into an improper fraction would give us 15 over4 so how do we do that what we would do is we would multiply our 3 by our denominator of four that would give us 12 and then we're going to add our three that is in our Nu our numerator giving us 15 over 4 what happens when we encounter repeating decimals so consider one of the most well-known repeating decimal 0.333 3333 which continues indefinitely we signify this reputation by adding and placing a bar over the number of three indicating 0.3 is a repeating decimal there are countless other examples of repeating decimals but the key takeaway is that any decimal with a repeating sequence no matter how long can be represented as a fraction this is going to hold true even with sequences with Mill millions of digits that repeat endlessly so you might be wondering at this point hey nurse Chun with everything that we've covered so far including integers finite non-re repeating decimals and repeating decimals what's left are there any numbers that don't fit into this category of rational numbers some of the most renowned numbers in the entire field of mathematics fall into this category and we call them irrational numbers these numbers cannot be expressed as a simple Ratio or two integers I've highlighted a section of significant examples that you're probably going to see on your t's so take Pi for example the ratio of a circle circumference to its diameter is known as an irrational number its digits extend indefinitely without termination and without forming any repeating patterns similarly e which arises from concepts of continuously compounding interests in complex analysis never ends or repeats itself also making it an irrational number the square OT of two is also irrational you might wonder if these examples are merely exceptions or perhaps most numbers are rational with only a few rare irrational cases that we're highlighting here however it's crucial to recognize that while these numbers might appear to be exotic due to their unique properties and applications they are not as rare as you might think consider this perspective I mentioned the square root of two as an example but if you calculate the square root of any number that isn't a perfect square you're going to find yourself with an irrational number moreover when you combine any irrational number with a rational number whether it's an addition or multiplication it's going to result in some kind of irrational number this demonstrates just how abundant irrational numbers actually are when we're looking at the mathematical landscape let's take a look at our first example which of the following numbers is irrational so let's break each one of the down so we have 8 / 2 is going to give us 4 we have the square Ro T of 16 also gives us four so we can automatically eliminate A and B as being the correct answer because remember we're looking for an irrational number next we have the square root of 15 oh look at that we have a very long number right we have 3.8 7 2 9 8 so on and so forth but let's finish out we have 2 point 75 2.75 is a rational number so out of all of the options that we have available to us C is going to be the most correct number because there's a lot of repeating decimals but they don't follow a pattern our next question states which expression results in a irrational number so again we're looking for irrational so starting off with our handy dandy calculator we have 25 / 5 is going to give us five so we know that that is rational we can eliminate that 2 raed to the^ 2 is 4 that's also again a rational number so we can eliminate that we have the square root of 49 the square root of 49 gives us 7 so again we know that that is also rational we can eliminate that and our last number is the square root of three and again look at that we have a very long number we've got 1.73 2 0 five Z so on and so on and so forth so out of all the options that we have available to us D is the most correct answer because remember we have a non-terminating decimal with no repeating patterns so when we see those kind of numbers we know we have an irrational number now that we have a better understanding of rational and irrational numbers let's discuss how we're going to order and compare rational numbers we're going to sequence these six numbers from smallest to greatest for the smallest number positioned all the way to the left of our number line and the largest number all the way to the right I encourage you to pause this video and try to arrange them yourselves before we tackle it together let's proceed with how we're going to get the solution once you've given it a shot we'll utilize a number line to assist us with this task which I prepared for your convenience let's begin by examining each number one at a time so when it comes to your te's the best option is to plot one option at a time and convert any fraction that you have to decimals to make it easier to compare on the number line so let's examine this process so we start with the number -3 over 4 so when we do our conversion -3 over4 is going to give us 0.75 so we're going to mark it on our graph right here next we have 0.5 0.5 is going to be more on the right side of our number line next we have -1.5 again looking at our number line it's going to be placed right in here 2 over three so again we're going to have to do a little math 2 / 3 is going to give us 0.6 and again we add our little bar on top of our six because it is indefinite six number pattern it's just 6 six 66 66 six so on and so forth down the line so 2 over 3 is going to be right about here on our number line next we have - one2 so - one2 is going to give us 0.5 and we are going to put that right here on our number line and then our last number is 1.75 so 1.75 is going to be way over here to the right of our number line so we're taking a look at this we've already plotted it we can get our numbers from least to greatest so least is all the way over here on the left and greatest is all the way over here on the right so we're just going to move our way down so I have the number I have the answer down here for you already so we have -1.5 34 12 0.5 2 over3 and 1.75 that is the correct answer for least to greatest with this numerical data set so we just learned about the numberline method when it comes to ordering and comparing rational numbers but there's a lot of people that also use the stacking method so I did want to address it in this video so the question is asking order the following set of numbers from least to greatest as we know when we're looking at a number line when it comes to the negatives the farther away that we get from zero the more negative it becomes right so what we do in this method is we're going to stack all the numbers vertically and line them up based on their decimal so if you look here all of the decimals are lined up so that we can compare our numbers now the reason why this works so well is I want to draw your attention up here to the right of your screen so if we're looking at a set of numbers that are lined up by their decimal points it really helps us understand how each number is greater than or less than so looking at our two examples we have 5.34 7 and 5582 so we have to figure out which one is greater which one is least so we start with the first number to the left of our decimal so they're both five so we automatically those are equal to each other we can move on to the right so our next set of numbers we have three and we have five well is three greater than five no absolutely not right so we can say that 5.82 is greater than 5347 so it just helps kind of visualize a little bit better if this method works for you so let's take a look back at our example here so again whenever I'm doing calculations with comparing and ordering rational numbers I always want to start with our negative numbers so I'm only going to focus on our 0.75 and our - 1.5 and our .5 so let's take a look so we start with our first numbers we have zero 1 and zero so as we discussed as we move further away from zero the more negative the number becomes and it also becomes the least So based on this we have -1.5 so we know that our first number our least number out of this data set is going to be 1.5 so next we're going to move on to our next numbers because both zeros are equal we can automatically eliminate those so now we have seven and we have five of our negative numbers again the further we we move away from the zero on the negative side of our number line the more least it becomes so 7 is greater than five which means that 0.75 is going to be less than 0.5 so now we've moved through all of our negative numbers we can automatically eliminate those now let's take a look at our positive numbers so with our positive numbers again we're going to be looking at the first number so we have 0 0 and 1 so one is obviously greater remember the further we move to the right of our number line the more greater the number becomes so we know that 1.75 is going to be our greatest number now we have to figure out which of these two remaining numbers are going to be greater than so we have five and we have six is five greater than six absolutely not so we can put 0.5 here and 0.6 here so out of our data set this is going to be our least to greatest all right our practice questions States arrange the following rational numbers from least to greatest so as we know based on our number line as we move further to the left it becomes more least as we move further to the right it becomes more great so let's take a look at each individual number so our first number we have is -34 well we know -34 is equal to 0.75 so we can automatically put that right about here and I always just write the number on top so that way I know let's take a look at our next number we have 0.5 we can go ahead and put that there we've got 1.25 so 1.25 next we have 2/3 so as we know 2 / 3 is equal to 0.6 with a bar above it because again it goes indefinitely to six in the same pattern so 0.6 is just going to be Aidy over here to the right of our 0.5 we'll put two over three and then lastly we have the number 1.2 so our 1.2 is going to go right about here 1.2 So based on the numbers we have we know the correct answer has to be -1.25 34 0.5 2/3 and 1.2 do we have any of these answers that make sense based on what we know yes we do the correct answer is going to be a something else the teas is going to test you on is comparing numbers using greater than less than and equal to when it comes to your rational numbers when we perform Solutions like inequalities we're going to start to see things like less than or equal to greater than or equal to as well but in this particular cases we're only focusing on three of them less than greater than and equal to here's a quick tip that I want you to remember when you're taking your t's when it comes to less than or greater than and remembering which sign you need to use I want want you to think of a very hungry alligator or Hungry Hungry Hippo or maybe even Pac-Man they're always going to want to eat the larger number because I live in Florida I'm going to use the alligator analogy so the alligator is going to want to eat the greater number so we have three and we have five here as an example so is three greater than five absolutely not right three is less than five so our little Gator here is wanting to eat that number five because it's very very hungry so we're we're going to use the less than sign right his mouth is going towards the higher number so let's take a look at our numbers so we're comparing 0.75 to 34s so when we look at our number line we have 0.75 right about here and we divide 3 by 4 it's also going to give us 0.75 so in this case we can say that 0.75 is equal to 34s our next example we have -1 1 t0 and 25 so - 1 1110 is going to give us about - 1.1 and 25ths is going to give us about 2.2 so is our -1.1 greater than 2.2 absolutely not right so we can say that -110 is less than 25s now looking at our last example we have 1.2 and 0.8 so 1.2 two we're going to plot her about right here and 0.8 is going to go right about here so it's 1.2 greater than 0.8 absolutely so we're going to go ahead and use the greater than sign as we know that 1.2 is greater than 0.8 so our question states which of the following is not true so we're looking for the one that's not true so we're going to break each one of these down so we have 34 is greater than 0.75 well if we convert 34s to a decimal that gives us 0.75 0.75 is 34s greater than 0.75 absolutely not right we know that this one is not true but let's go ahead and move through everything else so next we have -2 is greater than 0.6 so using our number line we know that - one2 when we convert it to a DE decimal is 0.5 and we have 0.6 so 0.5 would be here 0.6 would be right about here so is -2 greater than 0.6 yes absolutely so we can automatically eliminate that answer so next we have 0.5 is greater than 1/3 so as we know we have 0.5 and then 1/3 is going to be 0.3 with a little line of above it so is 0.5 greater than 1/3 yes absolutely so we can automatically eliminate that and then our last is 2.5 is greater than 5/3 so when we convert 5/3 we get 1.66 so on and so forth down the line and we have 2.5 so 2.5 would be about right here 1.6 would be right here so is 2.5 greater than 5/3 yes absolutely so the correct answer is a 34s is not greater than 0.75 I hope that this video is helpful in understanding arithmetic numbers when it comes to the aits as always if you need any additional assistance make sure you leave some comments down below I love answering your questions head of our nurse chunk store.com where there's a bunch of additional resources to help you be successful on your te's and as always I'll catch you in the next video bye