look at the numbers below identify which
numbers are rational and which are
irrational so just as a bit of review
just let's review what it even means to
be a rational number so this means that
you can be represented as the ratio
of
two integers the ratio of two integers
and if you're irrational
by definition it just means that you
you're not rational that you can't be
represented by as the ratio of two
integers so not
not
not rational so there's some low hanging
fruit over here because some of these
numbers are actually represented as the
ratio of two integers i'll circle those
194 is an integer 23 045 an integer
clearly the ratio of two integers this
fraction right over here so it is
rational
one is an integer eight is an integer
clearly the ratio of two integers so
this right over here is rational as well
the other ones are a little bit less
clear here i have 3.66
and then this dot dot dot i'm assuming
means that it's going to keep repeating
so it's three point sixes and the six
keep the sixes keep going more typically
or often you'll see this represented as
3.6 with a line over just the 6 to say
this keeps repeating forever so first
i'm just going to tell you a rule a
rational number is a ratio of two
integers and any number that can be
represented as a repeating
repeating
or
terminating decimal
terminating
decimal
can be represented as the ratio of two
integers and therefore is rational so if
you believe what i just told you that
repeating decimals can be represented as
it as as the ratio of integers then
3.666 and so on and so forth the sixes
keep repeating this right over here is a
rational number
and there's a more systematic way of
converting repeating decimals into
rational numbers the way i do it
involves a little bit of algebra but
just to give you to show you that this
can be i just want to give you an
example
if i divide 2 over if i divide 2 by 3
what does that equal to 3 goes into 2
no times but it goes into 20
it goes into 20 6 times 6 times 3 is 18
you then have a remainder of 2. bring
down another zero which is implicitly
there
three goes into twenty six times i think
you see a pattern here six times three
is eighteen subtract you'd get a two
bring down another zero so two thirds
two two-thirds is equal to
0.66 and it keeps on going which is
equal to 0.6 repeating
so this thing over here 3.66666 we could
write this as 3
plus two-thirds or three plus point six
six six six six six or three and
two-thirds and now this is just a mixed
number if you represent this as an
improper fraction you then have the
ratio of two integers so three times
three is nine plus two is eleven
eleven over three is the same thing as
three point six six six six six six six
six so on and so forth so i showed you
that it actually can be represented as
the ratio of two integers
the twelve times the square root of
eleven
cannot be and the simple way to
recognize that is it is the square root
of a of a prime number and the square
root of a prime number will always be
irrational square root
of
a prime
and you can try it on a calculator if
you like taking the square root of a
prime number
but you will see that you will get a
non-terminating
non-repeating decimal and so this right
over here i'll do the irrational ones in
orange
this right over here is irrational
negative
81.25 once again very easy to represent
this as
very easy to represent this as a mixed
number if you want it would be negative
81 and 25 over 100 which is the same
thing as one-fourth and then very easy
to represent this as an improper
fraction four times
negative eighty-one four times
eighty-one is three hundred and
twenty-four so it's negative three
hundred
and then another negative one so it's
negative 300 negative
325 over 4. if that confused you that i
i multiplied and then i had a negative
1. remember this negative applies to
everything here it's negative 81
plus negative 1 4 is how you should
think about it or you could just think
of it how would i convert 81 and 1 4 4
times 81 is 324 plus 1 was 325 and then
you have a negative out front but the
main focus of this is look clearly you
can represent a terminating decimal very
easily as as a ratio of two integers and
so this right over here is that's a
different shade of yellow this right
over here is clearly clearly rational
i'm doing the rationals in yellow and
i'm doing the irrationals in orange
square to 3 we have the square root of a
prime number it's going to be irrational
1.09
terminating decimal this is 1 in 97 over
a thousand or 97 thousandths so clearly
clearly rational
and then pi
pi and you'll see other numbers like pi
like e these are by definition
or by calculation or however you want to
think about it these are irrational
numbers 3.15 1
3.14159 and it keeps on going forever it
is non-terminating and it is
non-repeating so it is an irrational
it is an irrational number