An Intro to Rational and Irrational Numbers | Math with Mr. J

[Music] welcome to math with Mr J in this video I'm going to go through an introduction to rational numbers and irrational numbers we'll start with rational numbers and then move on to irrational numbers now simply put rational numbers are numbers that can be written as a fraction specifically a fraction of two integers so the numerator must be an integer and the denominator must be an integer now fractions can be written as decimals so anything in decimal form that either terminates so cuts off or repeats so continues on forever in some type of pattern those are rational numbers this may all seem confusing at first but as we go through our examples this will all make a lot more sense so any number that can be written as a fraction of two integers so the numerator must be an integer and the denominator must be an integer that's going to be a rational number now remember integers include positive and negative numbers and zero decimals and fractions are not included now one thing I do want to mention the denominator cannot be zero because that will give us a value that is undefined that's why Zer crossed off for the denominator so something to keep in mind let's jump into our examples starting with number one where we have six so we have a whole number here can we write six as a fraction of two integers yes for example we can write 6 over 1 that equals 6 or 12 over 2 that equals 6 36 over 6 that equals 6 all of those examples equal six so six can be written as a fraction of two integers so it's a rational number and all whole numbers are going to be rational what about -6 what about a negative number can we write this as a fraction of two integers yes -6 over1 = -6 or -83 = -6 so so -6 is a rational number as well let's move on to number two where we have a decimal 710 is this rational well it is a terminating decimal it cuts off so right away we can see that this is rational but can it be written as a fraction of two integers yes 710 can be written as 7 over 10 that equals 7/10 or 21 over 30 = 710 or even 70 over 100 = 710 so 7/10 can be written as a fraction of two integers so it's rational and just like we talked about number one a negative so 710 that's rational as well we can write this as 710 so any terminating decimals are going to be rational let's move on to number three where we have 2 and 75 hundredths similar to number two we have a terminating decimal so right away we can tell that this is rational but how do we write this as a fraction of two integers well we can write this as 275 over 100 or we can even use the simplified version of of that fraction 11 over4 so both of those are examples of 2 and 7500s written as a fraction of two integers now just like numbers 1 and 2 -2 and 7500s is rational as well we can write that as a fraction of two integers let's move on to number four where we have zero is zero rational yes we can write zero as a fraction of two integers 0 over 1 equal 0 0 over 25 equals 0 0 over 100 = 0 so yes we can write 0 as a fraction of two integers zero is rational now like I mentioned earlier in the video If we have zero in the denominator though so let's just say 1 over Zer this does not equal zero zero in the denominator gives us a value that is undefined so that's not going to be rational so again something to keep in mind let's move on to number five where we have a repeating decimal so 0.3 repeating this bar right here this means we have 0.3 and this three continues on forever so is this rational our repeating decimals rational can we write this as a fraction of two integers yes we can actually write any repeating decimal as a fraction of two integers for example a fraction of two integers that equals 0.3 repeating well 1/3 or 3 9ths so any repeating decimals are rational even something that has multiple digits that repeat will be rational so for example something like 0.181 1818 and this continues on so 18 repeating we have a pattern here we have something that repeats so this is rational fractional form let's come below here so 0.18 repeating we can do 18 over 99 or even two over 11 next for number six we have 1/4 well this is already a fraction the numerator is one which is an integer and the denominator is four which is an integer so this is rational now let's look at the decimal form as well 1/4 equal 2500s which is a terminating decimal so we can see that this is a rational number that way as well let's move on to number seven where we have the square OT of 25 well the square root of 25 that equals 5 so with the square root of 25 giving us the value of five is five rational yes five is a whole number and we can write it as a fraction of two integers 5 over 1 50 over 10 so on and so for so the square OT of 25 is rational lastly let's move on to number eight and at this point you may be thinking well what's an example of something that's not rational well the Square < t of 3 gives us a value that is not rational if we plug this into a calculator we get 1.7 3 2 0 5 and this continues on on forever now this decimal does not repeat it doesn't have a pattern and again it continues on forever so the decimal form doesn't terminate or repeat so there is no way for us to write this as a fraction of two integers so this is not rational again there's no way for us to write this as a fraction of two integers but you may be thinking can't we just write this as the < TK of 3 over 1 that's a fraction yes it is a fraction but remember a rational number is any number that can be written as a fraction of two integers and this numerator is not an integer again the numerator and denominator have to be an integer for it to be rational the square root of three is not rational it's what we call irrational we cannot write it as a fraction of two integers and go going off of this anytime we take the square root of something that is not a perfect square or a cube root of something that is not a perfect Cube and so on we get an irrational number and there are other examples of irrational numbers as well we will go over those in another video that link is down in the description so as far as rational numbers go if we can write it as a fraction of two integers then that number is rational if the decimal form either terminates or repeats that also shows that we have a rational number there's a basic overview of rational numbers let's move on to irrational numbers so here is our irrational numbers section but before we move on to irrational numbers let's do a quick recap of rational numbers just to make sure we have those down so taking a look at number one we have 12 9/10 and 0.2 repeating these are all rational numbers we can write them as fractions fractions of two integers and the decimal form of a rational number either terminates or repeats so for 12 how can we write this as a fraction of two integers well we can write this as 12 over 1 that equals 12 about 24 over two that equals 12 and so on so we can write 12 as a fraction of two integers so it is rational and the same thing goes for -12 so it doesn't matter if we have a negative this is still rational we can write this as -12 over 1 and so on so -12 is also rational let's move on to 9/10 now this is a terminating decimal so we can see that this is rational right away but as as far as writing this as a fraction of two integers we can write this as 9/ 10 or even something like 36 over 40 and so on so we can write this as a fraction of two integers 9/10 is rational and then lastly we have 0.2 repeating this bar over the two means we have 0.2 2 2 2 and and this continues on forever now any repeating decimal is rational it can be written as a fraction of two integers for example 0.2 repeating can be written as 2 over 9 or 6 over 27 and so on let's move on to irrational numbers and we will start with the square OT of two now the square root of two is approximately so let's put approximately and I say approximately because this decimal does not terminate or repeat so we can't write out the full decimal so the square < TK of two is approximately 1.44 2 1 3 5 6 2 and this continues on again it does not terminate or repeat so this is irrational there's no way to write this as a fraction of two integers now you may be thinking well let's just write it as the square < TK of 2 over one yes it is a fraction but it has to be a fraction of two integers the square root of two is not an integer so again we cannot write the square root of two as a fraction of two integers and the decimal doesn't terminate or repeat this is actually going to be the case for taking the square root of anything that is not a perfect square so the square root of any non-perfect square is irrational let's move on to number three where we have 4.89 8979 48 and then that decimal continues on it does not terminate or repeat so this is irrational we cannot write this as a fraction of two integers now this decimal is actually what we get when we plug in the square root of 24 into a calculator next for number four we have Pi which is a very famous irrational number that we use when working with circles Pi is the ratio of a circle's circumference to its diameter so the circumference divided by the diameter Pi is the same for every circle now Pi is approximately 3.14 1 [Music] 5 9 2 6 and this continues on forever without repeating now we cannot write pi as a fraction of two integers Pi is irrational now it's very common to use 3.14 for pi but that's just an approximation let's move on to number five where we have the number e also called Oilers number named after the mathemati Leonard Oiler this is another famous irrational number that shows up throughout math with things like exponential growth compound interest probability among other applications the number e is approximately 2.718 288 2845 and this continues on forever ever without repeating so we can't write the number e as a fraction of two integers it's irrational now let's move on to numbers 6 through 10 where we will determine if we have something that is rational or irrational for number six we have -63 this is rational we can write this as a fraction of two integers for example -63 over 1 so 63 over 1 again this is rational next for number seven we have 31723 and that decimal continues on without terminating or repeating this is irrational we can't write it as a fraction of two integers this is actually the decimal that we get for the cube root of 30 so this is irrational next let's move on to number eight where we have the square root of 49 Now 49 is a perfect square so the Square < t of 49 equals 7 so the square < TK of 49 rational moving on to number nine we have 8. 71425 now above 71425 we have a bar that shows us that that decimal repeats and again any repeating decimal is rational so this is rational lastly let's move on to number 10 where we have the square root of 50 50 is not a perfect square so this is not going to work out to be a rational number the square root of 50 is approximately 7.07 1 06 and this continues on forever without terminating or repeating so the square root of 50 is irrational so there you have it there's an introduction to rational numbers and irrational numbers I hope that helped thanks so much for watching until next time peace