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Understanding Real Numbers: Rational vs Irrational Made Easy!
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May 11, 2025
In this Pre-Algebra lesson, I break down the entire real number system—from rational and irrational numbers to natural numbers, whole numbers, integers, and more! You'll learn: The difference between rational and irrational numbers (with clear examples) How to classify numbers like √5, -2.5, 21/7, and π What makes a number “real” and how real numbers are organized Why some equations have no solution in the realm of real numbers A sneak peek into complex numbers (coming next year!) Perfect for middle schoolers, high schoolers, and anyone needing a refresher on real numbers.
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0:01
i said there's enemies now all right so
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we're going to talk about real numbers i
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don't even know why I suggested that we
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should talk about it we need to talk
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thank you so we're going to talk about
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real numbers okay but before we do that
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in the beginning of the year we talked
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about what's the word rational numbers
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and all these things so today we're
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going to go into depth right i got you
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so before we talk about real numbers I
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want to introduce you to irrational
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numbers right irrational when I say
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something is irrational what am I
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meaning by that the word irrational what
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does that mean that it's not doesn't
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make sense right irrational right so
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rational numbers are numbers that can be
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written as a fraction right an example
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of a rational number would be 1/2 right
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1/2 is a rational number rational
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right rational
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numbers those can be written or
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expressed as a fraction right we did
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rational numbers one over two is a
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rational number right what else 0.5 uh
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0.25 is a rational number why is he a
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rational number cuz you can write 0.25
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as a fraction we did that you all
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remember that how do you write 0.25 as a
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fraction
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what numbers no 1/4 1/4 well but before
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we did that we did what 25 over 100
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right we we move the decimal place how
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many places to the left to the right how
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many places four two places and we put
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it under what 100 right so 0.25 is 25
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over 100 and if you simplify that it
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gives you 1/4 these are all rational
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numbers right we also had a number like
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1 6
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0.33 that's also a rational number these
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are all rational numbers now we're going
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to talk about irrational numbers
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irrational numbers are numbers like pi
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right pi pi is an irrational number
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because if you put pi in your calculator
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you're going to get like
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3.14159 and a bunch of other numbers
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that never end right so therefore we
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refer to that number
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hey we refer to that number as a
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irrational number right also square root
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of two is an irrational number because
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if you put square root of two in your
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calculator you're going to get
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1.414 and then a bunch of other numbers
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that's also an irrational number square
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root of five is also an irrational
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number because if you put in your
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calculator you're going to get a string
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of different numbers that never end so
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those are called irrational numbers and
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another thing about irrational numbers
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is you cannot express them as a fraction
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you can't write 5 as a over b meaning
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you can't turn this into a fraction
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therefore they are called irrational
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numbers right now
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uh real numbers is almost like the
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parent of all numbers right after real
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numbers in next year if you take algebra
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one or two you're going to do complex
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numbers right
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i²=1 those are complex but right now
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we're not going to talk about complex
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but real numbers is right now we're
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gonna call real numbers the big daddy
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right big daddy's real numbers and big
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daddy
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excuse me so real numbers are going to
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be like the big papa or whatever and
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then he has children right and the
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children of the real numbers are a bunch
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of them so real numbers uh and compass
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rational numbers irrational numbers
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natural numbers whole numbers
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integers and then rational and
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irrational right so natural numbers they
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begin at what I learn how to do my taxes
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where where do natural numbers begin
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number one
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right
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i'm I'm I'm going to start talking when
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you guys be quiet okay
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you too that's enough
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so we have natural numbers natural
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numbers begin at one and they go up to
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infinity right natural numbers
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are 1 2 3 4 5 6 7 8 9 10 11 all the way
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to infinity those are natural numbers
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whole numbers begin at zero right whole
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numbers are 0 1 2 3 4 5 6 7 8 9 10 to
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infinity so the difference between
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natural and whole numbers is that
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natural numbers do not include zero but
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whole numbers begin at zero you
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understand right yeah what's the number
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before infinity there's no number before
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infinity girl and how can you go to
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infinity you can go to infinity because
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it's limitless the biggest number you
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get you add one to you get another so
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this why negative infinity to positive
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infinity
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when
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they do you never stop you going to keep
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going it doesn't stop you know how like
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you count 1 2 3 4 5 6 7 8 9 like you can
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literally like keep on adding numbers to
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your like pattern statements and forever
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they will never stop right negative
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infinity to positive infinity you're
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never going to get an end number so
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there's no such thing as the highest
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number that ever existed right because
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any number you get you can get something
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higher than that yes you make samples of
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real numbers yeah we're going to talk
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about it right
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so all numbers that we deal with are
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real numbers right for example
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Mr
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okay that's funny
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we'll do we do some other thing we
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forget sometimes right so here's the
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thing natural numbers
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why am I hearing voices all day today
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please keep your mouth shut so natural
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numbers begin at 1 1 2 3 4 5 to infinity
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whole numbers begin at 0 0 1 2 3 4 5 to
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infinity integers include negative
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numbers and positive numbers integers
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are like5 -10 -2 -3 1 2 3 0 and all of
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that right so if you see the progression
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of it first you have natural you have
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whole then you have integers and then
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you have rational rational numbers are
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all the numbers that can be written as
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fractions right and then the last one is
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irrational now irrational numbers are
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the ones that I gave you just now and
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all of those make up what we call the
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real numbers so a real number can be
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classified as every single one of those
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so real numbers include natural whole
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inteious rational and irrational and
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then decimal numbers as well yeah are
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there any numbers that aren't a real
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number yeah this number here 2 - 3 I
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that's called a complex number you know
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you're going to see that
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so this is called time complex one I
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complex number right what does complex
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number mean it's a complex it's a number
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that can be written in a form a plus b I
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call it's called an imaginary number
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right imaginary so complex numbers are
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going to be called erase that we're not
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doing that you're going to do that next
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year trust me
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Oh that's next year that's still a year
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all right so we're going to see complex
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number next year but right now we are
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sticking with real numbers all right so
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now I'm going to give you we're going to
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learn how to classify numbers right so
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number 21 over 7
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too much talking over here yeah the
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homework is not in class this is a
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lecture now right so 21
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so what kind of number is that sure so
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21 over 7 what does that give you really
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three right 21 is equal to three so you
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quite fast how many numbers where can
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you put three um natural whole integer
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right if it's natural it is right it's
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also whole is also what an integer it's
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also a rational and it's pretty much a
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real number right this is a real number
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it's a real number to begin with and
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it's all of these right is natural is
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whole is an integer and it's also
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rational because you can write this as
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three so basically everything other than
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irrational then they're irrational
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because 21 over 7 is three natural
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numbers begin at zero no begin at one
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whole numbers begin at zero and then go
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to infinity integers includes negative
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numbers and positive rational can be
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written as fractions right three can be
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written as a fraction three is what
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three over one so therefore is a
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rational number now what about -2.5
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can it be a natural number no can it be
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natural no yes no are you sure can it be
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natural no yes no can it be natural
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naturals are what
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can it be a whole number no can it be an
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integer no it can't what just be
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negative and positive numbers negative
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and positive integers -23 this is -2.5
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he has a decimal 2.5 so therefore it's
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not an integer can he be irrational yes
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because you can write this as what a
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fraction - 255 over 10 right this can be
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written as - 255 over 10 so therefore
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this is a rational number right it's a
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rational number what about 0.2 with a
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thing on top of it what kind of number
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is that irrational it's not irrational
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it's going to be rational because you
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can write this as a fraction remember we
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did that rational that's irrational any
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number repeating with repeating digits
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like that can be called irrational
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number so that's a rational do that
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again why would you do that do that
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later on but we did how would you
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multiply by 10 x and then we did all of
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that how would you 0.2 in a fraction you
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can you can you start by saying 0.2 no I
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know i'm just And then remember how we
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did it we multiply we let x equals this
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right and then we did what 10 x= to what
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two right 10 x= 2 and we did two minus
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that and then we find it to be uh 9 over
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yeah i don't I don't remember two we'll
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do that later i don't want I don't want
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to get into that right now so Oh that
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was after I added yeah that's like in
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the beginning of the year right now
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square root of 38 what kind of number is
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this irrational irrational irrational
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because is it a perfect square no no
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right it's not a perfect square so this
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is an irrational number if you do square
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root of 38 if you put in your calculator
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you're going to see that's going to give
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you a string of like random numbers
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going all the way to like whatever so
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that is an that's an irrational number
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what about 9 over five rational rational
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square root of 100
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[Music]
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irrational
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what's 100 10 10 so what is 10 natural
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is this here is also whole is also
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integer is also rational right cuz
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square root of 100 is a perfect square
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so therefore you can find a whole bunch
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of stuff it can be a natural it could be
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a whole it could be an integer it could
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be all of that right what about square
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root of
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u four what's that
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two rational all right what about what
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about square root of uh Um square root
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of 16
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four except everything irrational what
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is it
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16 is irrational the square of 16 is
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four the square root of four is two so
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therefore that is a natural it's a
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natural is a whole is an integer count
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like that right what about uh double
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let's do this here what about uh square
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root of
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um
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is it natural because it's a whole
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number 900 900 oh what is it um
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natural because it's still what
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30 so therefore it's natural is
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is also rational right so that order of
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that okay all right now so we just
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learned how to classify numbers so now
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we're going to learn how to compare real
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numbers right all of these are real
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numbers an irrational number is also a
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real number we already know because like
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I said the real number the set of real
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numbers encompass all of these right so
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you can have your own set you can create
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your own set right you know how people
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say throw your set in the air what's
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your set what do you mean what does that
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mean like a ben
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what's your set what's your set it's
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like a gang or something no not a
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gangster
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[Music]
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my set is what my set is this what is
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this
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you tell call yourself a loser
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what is your set what what is it this is
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love and this is
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Jesus jesus love you could still love
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each other yeah give me that right
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i don't know about you
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all right anyway
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all right so are you are
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you strong
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all right 3 and 1/3 and then square root
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of 15 right so let's try to figure out
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what this is so how do you compare these
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numbers so which one is bigger how we
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going to do it we going to turn this
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into what fraction fraction so what
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would I give you 10 10 over three right
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3 and 3 is 10 9 9 and 1 is 10 over 3
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right and then square root of 15 now
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what we have to do is this we have to
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divide 10 by three what do we get put on
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your calculator 10 over 3 tell me what
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you got 3.3
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what do you have 3.1 what tell me
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3.3 3.3 all right what about square root
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of 15 so put that in your calculator
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square of 15
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it's going to give you something weird
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uh 3.83
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89 what
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8729 83346
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3346 all right so therefore if you were
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to compare them which one would be less
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it's easy right this one is less than
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that because 3.3 is certainly less than
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3.8 so therefore 3 and 1/3 is less than
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square root of 15 this is really simple
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stuff right now we're going to talk
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about something bigger something more
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hard we're going to talk about solving
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equations with uh solving square
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equation right solving equation with
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squares all right so let's say I have
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this here
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i want to solve an equation what time
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does end 95 oh it's so
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solid this class always ends the same
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time it does yeah it always ends at 9:15
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or 910 all right I want to solve this
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right
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i think it's
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x= 36 right and I want to solve for x so
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what do I do
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where does 15 come from so x²= 36 and I
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want to solve for x to um is x2
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all right so x^ 2 is x * x all right so
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you just do the square root of 36 all
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right to
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find the square root right because
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here's the thing i say what number
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multiply by itself twice gives me 36 i
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have two options it's not just six what
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else 50 no it's not just six oh 9 * 4 no
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what number multiplied by itself gives
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you six so you take the square root
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right if you take the square root you're
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going to get two values x is going to be
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equal to what plus or minus
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six what does that even do can you put
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that in you know why why is it plus or
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minus plus or minus because here's the
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thing right
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6 square is equal to what
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12
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36
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and then -6 square is also equal to what
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36 no just 36 36 so therefore I have two
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solutions
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six I see that oh it could be what -6
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right but you have negative in in
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parentheses so you can have negative in
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parentheses so now if I give you another
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example if I say x² is equal to 25 so
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how do you solve for x here one number
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one multip by itself twice gives you
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plus what x = what plus
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Rus right
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now so far we have perfect squares now
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we're going to have something that's not
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a perfect square so how we solve it so
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what if I have erase that what if I have
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now
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right what if I have
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2x² is = to let's say um 10 and I'm
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trying to solve this divide by two first
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what do we do divide by what
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i want to get x by itself right okay so
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so I'm going to divide by two two right
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so I get x² = to 5 mhm and then what do
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I do here i'm going to take the square
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root right but now it's a prime number
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though five is not perfect square so
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it's going to be x= what orus
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what we leave like we leave it like this
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because we know that cannot be found
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right square of 5 is not a perfect
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square so we're going to leave it as
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plus or
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minus 5 right now let me trick you
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here see if you can catch it x² is
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=16 how do you solve this
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x
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you can't because that's a negative you
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can't square a negative hey you got it
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good job we can't have a any anytime you
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square a number you're never going to
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get what a negative
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so this is impossible cannot be solved
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in the realm of the real numbers but
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next year when you do complex numbers
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we're going to go
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x²= 16 I and then x be for next year for
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I right next year we going to do this
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are we doing this next year you were
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going to do this in Mrs up
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in one honor no
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