[Music] welcome to math with Mr J in this video I'm going to go through a mini course on integers we will start with what integers are then move to comparing and ordering integers and end with adding subtracting multiplying and dividing integers so whether you are new to integers using this as a review or whatever the case may be this will give us a nice over viiew of working with integers so let's begin by taking a look at what integers are now simply put integers are numbers with no fractional or decimal part integers include positive numbers negative numbers and zero let's go through some examples to better understand this starting with number one where we have a number line we're going to start at zero which again is an integer to the right of zero we are counting up by one we are increasing in value so 1 2 3 4 5 so on and so forth and those numbers continue on forever they are infinite those are all of our positive integers to the left of zero we are counting down by one we're decreasing in value so so -1 -2 -3 -4 -5 so on and so forth those numbers continue on forever as well they are infinite those are our negative integers going back to zero zero separates the positives and negatives zero itself is not positive or negative moving on to number two we have the integers written as a set so we have the integers listed between those braces so this is just a different way to look at integers other than that number line above we can see Zero right here with the positive integers to the right and those three dots tell us that those continue on and then the negative integers are to the left of zero with again those three dots showing us that those continue on as well so really integers are just all of the numbers counting up by one from zero those are the positives and then all of the numbers counting down by one from zero those are the negatives and then zero as well let's move on to number three and write out a few more examples of integers let's start with positive integers some examples include 8 37 and 194 now let's write out a few examples of negative integers -9 -5 and 72 and then lastly we know zero is an integer and 0 is just zero moving on to number four let's write out some examples of numbers that aren't integers so for examp example 1/2 is not an integer 4 and 38 is not an integer 6 and 310 6.3 is not an integer and 45 hundredths 0.45 is not an integer and we can see that all of those examples include a fractional or decimal part lastly for number five let's mark yes or no if the number is an integer so we will start with a 42 so is 42 an integer yes next for B we have -8 and 510 -8.5 we have a decimal part involved here so no this is not an integer next for C we have 3 and 1/3 so there's a fractional part here so no this is not an integer next for D we have zero which zero is an integer moving over to e we have -7 is -7 an integer yes moving on to F we have1 12ths so is this an integer no we have a fractional part here next for G we have 996 is this an integer yes and then lastly H we have 10 and 75 hundredths 10.75 is this and integer no there's a decimal part here so there's what integers are let's move on to comparing and ordering now let's take a look at comparing and ordering integers we will start with comparing and then move on to ordering for comparing we will go through three examples where we will use a number line to help us compare and then we will move on to three more examples where we will not use a number line now when comparing integers it can be helpful to visualize the integers on a number line just remember the further right on a number line the greater the value numbers increase in value as we go right on a number line on the other hand the further left the smaller the value numbers decrease in value as we go left on a number line let's jump into number one where we have -1 is greater than less than or equal to -2 well let's take a look at the number line -1 is right here and -2 is right here so looking at that number line which integer is further to the right that means it's going to be greater in value -1 is further right -1 is greater than -2 so let's fill in our comparison here and remember we want the comparison symbol opening up towards the greater value so like this and we can read this comparison from left to right as -1 is greater than -2 now before we move on to number two I do want to mention a couple of things here when we compare two negatives the one closer to zero is going to be greater the one closer to zero is further to the right so that's just a slightly different way of thinking about it when we have two negatives and another thing I like to do when comparing integers is related to temperature and that can help us think through these for example is -1° fahren or Celsius here it doesn't matter the unit is is that warmer or colder than -2° -1 is warmer by 1 Dee it's a higher temperature than -2° so -1 is greater so there are different ways to think through these use what works best for you moving on to number two we have -5 is greater than less than or equal to 3 let's take a look at the number line three is right here and -5 is right here so which of those is further to the right well three 3 is greater than -5 So reading this comparison from left to right -5 is less than three now with number two we have a negative and a positive keep in mind whenever we have a negative and a positive the positive is always going to be greater than the negative and relating this to temperature which is going to be warmer which is going to be the greater temperature -5° or 3° 3° moving on to number three we have 0 is greater than less than or equal to -4 let's take a look at the number line 0 is right here and -4 is right here zero is further to the right on the number line therefore it's greater than -4 So reading this from left to right 0 is greater than -4 and to go off of this whenever we have zero and a negative 0 is always going to be greater than a negative let's move on to number four where we have 95 is greater than less than or equal to 6 well we have a negative and a positive here the positive is automatically greater than the negative and thinking in terms of a number line six is going to be further to the right than 95 So reading this from left to right 95 is less than 6 taking a look at number five now we have -14 is greater than less than or equal to11 so we have two negatives here which one is closer to zero which one is further to the right on a number line or if we're thinking temperature which is the warmer temperature however we want to think about it -1 is greater So reading this from left to right -4 is less than1 lastly taking a look at number six we have -27 is greater than less than or equal to -29 so we have two negatives again here so which one is closer to zero which one is further to the right on a number line that's going to be ne27 -27 is greater So reading this from left to right -27 is greater than -29 so there's how to compare integers let's move on to ordering integers now let's take a look at ordering integers we will go through two examples where we will use a number line to help us out and then we will move on to two more examples where we will not use a number line let's jump into number one where we need to order these integers from least to greatest we'll start by marking them on the number line zero is first which is right here then we have -2 then -1 then 4 and last ly -5 so again we want least to greatest here and remember the further left the smaller the value the further right the greater the value so the integer furthest to the left will be our least which we need to start with that's going to be5 so let's start with -5 and now we can go from there all we need to do is work our way right we increase in value as we go right so we're working our way from least to greatest so next is going to be -2 then -1 then zero and then lastly the greatest four and now we are in order from least to greatest we're done so neg5 -2 -1 0 and 4 now before moving on to number two I do want to mention just like when we compare integers I like to relate this to temperature to help us think through these -5° is the coldest temperature here and we work our way up to the warmest temperature 4° so just a different way to think through these let's move on to number two where we need to order these integers from greatest to least let's put these on the number line starting with three so three is right here next is -3 then we have 5 then 1 and then lastly -2 now we need to go from greatest to least so we need to start with the integer furthest to the right on the number line that's five so we will start with five and we need to work our way left now we decrease in value as we move left so we are going from greatest to least next is three then we have 1 then -2 and then lastly the least here3 and now we are in order from greatest to least 5 3 1 -2 and and-3 let's move on to numbers three and 4 taking a look at numbers 3 and four let's jump into number three here first where we need to order these from least to greatest so we need to start with the least so which of these is the furthest to the left on a number line or think which of these is the lowest or coldest temperature however we think about it that's going to be negative 19 that's the furthest to the left on the number line so our least -19 so we will start there and I'm actually going to cross -19 off in our original list in order to help us stay organized and focus on what's left now looking at what's left well we need to work our way to the greatest so what's the next lowest in value we have -10 20 20 6 and -11 so try to picture a number line here and we are working our way right from 91 is next so1 and I'll cross that off and now we have -10 20 and 6 so -10 is next and now we are down to the last two 20 and six well six is next and 20 is the greatest so six and then 20 so we have all of our integers there and we are in order from least to greatest -91 -10 6 and 20 lastly let's move on to number four where we need to order these from greatest to least so let's start with the greatest so which of these is furthest to the right on a number line we have8 17 8 10 -7 and --12 the greatest is 17 let's start there so 17 and cross that off so looking at what we have left8 8 10 -7 and -12 10 is going to come next that's actually the only positive number left so that's going to be after 17 here now we have8 -8 -7 and -12 so we are working our way left on a number line so next is 7 and now we are down to three 8 -8 and -12 next is -8 and we are down to two here -8 and --12 -12 is next that will bring us to our last integer here the least negative 18 and now we are in order from greatest to least 17 10 -78 -12 and8 so there's how to compare and order integers let's move on to adding subtracting multiplying and dividing integers now let's take a look at adding subtracting multiplying and dividing integers we will go through two examples for each we'll start with adding integers and when it comes to adding integers there are a couple of different ways to think through these so I'll show you two ways to go through each of our examples here let's jump into number one where we have 12 + -7 we'll start this problem by taking a look at the signs we have a positive2 and a -7 so we have different signs a positive and a negative since we have different signs we are going to take the greater absolute value and subtract the Lesser our answer will take the sign of the greater absolute value let's start by writing the absolute value of both 12 and -7 and remember absolute value is the distance a number is from zero the absolute value of 12 is 12 the absolute value of -7 is 7 now we take the greater absolute value and subtract the leser these are already in order so we can subtract if the larger absolute value comes second you can always switch the order to find the difference if need be let's subtract so 12 - 7 is five now we need to determine if our answer is going to be positive or negative so we need to take a look at the larger absolute value which is this 12 so we take the sign of the larger absolute value from the original problem so the larger absolute value is 12 let's take a look at the 12 in the original problem and that 12 is positive that means our answer is going to be positive so our final answer a positive five so a quick recap here basically we forgot about any negatives because we were working with absolute values we then found the difference between the absolute values and the answer takes the sign of the greater absolute value from the original problem now let's think through this another way and this way is going to be more of a mental math approach just basically thinking about what's going on in this problem so let me rewrite 12 plus -7 here so our original problem so we are starting at a positive2 and we are adding a -7 by adding a negative by adding that -7 we are decreasing in value by seven from that 12 we can basically think of this as 12 - 7 or 12 take away 7 that gives us our answer of five so 12 + -7 we are decreasing ining that 12 by a value of seven so we get a positive 5 so again we started at a positive2 always think about where you are starting and where you are going from that starting point so we are adding a -7 which is decreasing our 12in value by seven and we end up with five let's move on to number two where we have 8+ 10 here we have two negatives so the same signs so we're going to add the absolute values and use the same sign so let's start by taking a look at the absolute value of8 and -10 the absolute value of8 is 8 plus the absolute value of 10 which is 10 now we add those absolute values because again we had the same signs 8 + 10 is 18 we Ed the same sign from the original problem which those are negatives there so our answer is negative Final Answer 18 now if we were to think through this we can think that we are starting at8 so let me rewrite here8 Plus -10 so again starting at8 and we are adding a -10 so that means we are decreasing in value by 10 that leaves us at -8 like I mentioned earlier think about your starting point so the number you are starting with we have a -8 and then adding that -10 tells us we are decreasing in value and end up at8 -8 is our final answer that's how we add integers let's move on to subtraction so here are our examples for subtracting integers let's jump into number one where we have five - -9 now when we subtract integers we're actually going to add the opposite so if you're able to add integers you are going to be able to subtract the opposite of sub subraction is addition and then we take the opposite of the number we are subtracting so this gives us an equivalent problem and we are able to use this strategy so we have [Music] five and then let's add the opposite of -9 the opposite of --9 is positive 9 so 5 + 9 that gives us 14 a positive 14 and that's our final answer now that answer may not Mak sense at first but let's think about how we end up with a positive 14 in this subtraction problem whenever we subtract a negative we actually increase in value I like to think of this in terms of money a negative represents a debt or an expense when it comes to money so that -9 would be a $9 debt or expense think of subtracting a negative like subtracting or taking away that debt or expense and getting that money back that is a good positive thing and increases the value of that problem so something to think about let's move on to number two where we have -3 - 20 so let's add the opposite -3 plus the opposite of positive 20 is -20 so we have -3 + -20 now adding that -20 we are decreasing in value by 20 so we're starting at -3 and then decreasing in value by 20 that's going to give us 20 three and that's our final answer that's how we subtract integers let's move on to multiplication here are our examples for multiplying integers let's jump into number one where we have -7 * pos4 we have a negative * a positive so we are working with different Signs Now when we're working with different signs this tells us our product the answer to a multiplication problem will be negative so -7 * 4 well 7 * 4 is 28 and then we know this product is negative because we are working with different signs so this is NE 28 and that's our final answer to recap different signs equal a negative product so if we have a negative time a positive that will equal a negative product or if we have a positive time a negative that will equal a negative product let's move on to number two where we have ne10 * -6 so we have the same signs there we have two negative numbers so our product will be positive so let's think 10 * 6 is 60 and we have the same signs there so this is going to be a positive 60 our final answer 60 so to recap same signs equals a positive product soga * a negative equals a positive product and then a positive * a positive also equals a positive product that's how we multiply integers let's move on to division so here are our examples for dividing integers and you'll notice that our rules are the same for dividing integers as they were for multiplying integers as far as working with those positives and negatives and if we're working with the same signs or different signs let's jump into number one where we have -48 / -8 for now let's just think of this as 48 / 8 and what that quotient will be remember the quotient is the answer to a division problem so 48 divid 8 is 6 so let's write a six here and now we need to determine if our quotient is positive or negative well we had a negative divided by a negative so anytime we have the same signs in our original problem our quotient is positive so same signs positive quotient so -48 / -8 equal a 6 now let me back up here and further explain same signs so same signs would be a negative divided by a negative like number one that equals a positive and then also a positive divided a positive equals a positive let's move on to number two where we have 36 a positive 36 / -4 so let's think of this as 36 / 4 which is 9 now we need to determine if that quotient is positive or negative well we have a positive divided by a negative so we have different signs different signs that is a negative quotient so -9 there 36 / -4 equal -9 so for different signs that means a positive / a negative like number two that equals a negative and then a negative / a positive equals a negative so there you have it there's a basic overview of of working with integers I hope that helped thanks so much for watching until next time peace