Rational Numbers Explained | Math with Mr. J

[Music] welcome to math with Mr J in this video I'm going to cover what rational numbers are 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 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 zero is crossed off for the denominator so something to keep in mind let's jump into to 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 six 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 ne6 what about a negative number can we write this as a fraction of two integers yes ne6 over 1 = -6 or -8 over3 = -6 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 equals 710 so 710 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 7500s Sim 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 that fraction 11 over4 so both of those are examples of 2 and 7500 written as a fraction of two integers now just like numbers 1 and 2 -2 and 75 hundredths 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 = 0 0 over 100 = 0 so yes we can write 0 as a fraction of two integers Zer 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 0 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 03 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 0.1 1818 18 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 2 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 25 hundredths 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 root of 25 well the square OT 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 forth so the square < TK 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 < TK 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 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 square < 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 going off of this anytime we take the square root of something that is not a perfect square or a 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 so there you have it there's an explanation of what rational numbers are I hope that helped thanks so much for watching until next time peace