This video provides a comprehensive introduction to different types of real numbers, including natural, whole, integers, rational, and irrational numbers, explaining their definitions, relationships, and properties.
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hi friends if you watch this whole video
you'll find the topic real easy the
topic will seem natural to you so what's
the topic for today that's right this
video is about numbers in this video
we'll explore the different type of real
numbers such as natural numbers whole
numbers integers rational and irrational
numbers and then we'll finish off with
our top three test oriented questions on
this topic let's go back to the
kindergarten days and start with
counting numbers remember the Rhine one
two three four five once I caught a fish
alive well these numbers such as 1 2 3
and so on are called
natural numbers and they are denoted by
the letter M let's draw a circle to
represent the set of natural numbers now
whole numbers includes all the natural
numbers and zero so we can draw the
whole number circle around the natural
numbers integers include all the whole
numbers and negative numbers such as
minus 1 minus 2 minus 3 and so on so we
can draw the integer circle around the
whole numbers one important thing for
you to keep in mind is that zero is
considered neither positive nor negative
let's place the diagram we just did on
our concept board now the question is
are there any numbers between any two
integers let's say between 0 & 1 yes and
we use them all the time for example
half one-third point seven point eight
so between any two integers you have
these non integer numbers which are
known as fractions or decimals
and this entire set of integers and
fractions and decimals are called
rational numbers and they are denoted by
the letter Q any rational number can be
written as a fraction P by Q where P and
Q are integers the only restriction is
that Q should not be 0 since division by
zero is undefined now is the natural
number 3 also a rational number what do
you think so 3 can be written as 3 by 1
it can be written as a fraction you know
P by Q form so 3 is also a rational number
number
now what about 2.5 so we can write it as
25 by 10 and if you simplify the
fraction you get 5 by 2 so 2.5 is also a
rational number so any rational number
can be expressed in a P by Q form that
is in this fraction form the only
restriction is Q should not be equal to
0 and typically P and Q don't have any
common factor except 1 so we write the
fraction in its simplified form let's
add the non integers to our set diagram
the integers and non integers that is
fractions and decimals together are
called rational numbers let's talk more
about rational numbers that are not
integers as we learned rational numbers
are fractions represented by the P by Q
form you can calculate the fraction by
dividing and we end up with a decimal
number for example 7 by 10 is 0.75 by 2
is 2.5 7 by 4 is 1.75 these are all examples
examples
apples of terminating decimal numbers
that is the decimal digits stop or
terminate after a point because on
dividing by the denominator the
remainder is zero but there are other
fractions which on dividing don't
terminate for example two by three is
zero point six six six six and it just
goes on so we write it as zero point six
with a dot or a line on top to represent
that the digit six is recurring which
means repeating let's take another example
example
one by seven so if you look at the
answer here can you see the repeating
pattern that's right it's zero point one
four two eight five seven recurring
let's take a look at another example one
by six so that's going to be zero point
one six six six six and it just keeps
repeating but notice that we write it
with a dot or a bar only on top of six
because only the six is repeating so
these are called recurring decimal
numbers well on dividing by the
denominator the remainder is never zero
and the digits in the quotient keep
repeating if you take any fraction how
do you determine that on division you're
going to get a terminating decimal
number or a recurring decimal number of
course one simple way is to actually do
the division
but there's another trick where you
don't need to divide now what's the
trick you need to take a look at the
denominator and if the denominator has
factors of two and five only then you're
going to get a terminating decimal
number otherwise it's going to be a
recurring decimal number let me pull up
some examples to illustrate this let's
start with the example of seven by
for now what are the factors of the
denominator for its two into two now our
goal is to convert the denominator into
multiples of tens so I'm going to
multiply 5 into 5 in the numerator and
in the denominator so I get 175 by 100
that's 1.75 so this is a terminating
decimal number but if we take 37 by 150
so what are the factors of the denominator
denominator
they are 2 into 3 into 5 into 5 and if
you divide we get zero point 2 4 6 6 6
and so on so this is a recurring decimal
number and why is that because if you
carefully look at the factors in the
denominator you can see that there's a
three there for it to be a terminating
decimal number you need factors of two
and five only now our rational question
is are there numbers that are not
rational the answers yes
these are called irrational numbers for
example root 2 root 3 cube root of 10
these are all irrational numbers but we
look at these in a separate video
let's try to place the different numbers
that we've learnt on a number line are
you familiar with the number line an
everyday example is a ruler or a
measuring tape like this with numbers
marked on it unlike the measuring tape
the number line has both positive and
negative numbers so let's draw a number
line and try to place the different
numbers on it so here's our number line
let's mark the natural numbers on it 1 2
3 and so on let's draw another number
line and Mark the whole numbers on it so
it's going to be the natural numbers and
0 in our third number line let's mark
the integers
so it's going to be all the whole
numbers and the negative numbers minus 1
minus 2 minus 3 and so on in the next
number line let's mark the rational
numbers so it's going to be all the
integers and decimals and fractions for
example one point five is here midway
between one and two two point eight is
here and it's closer to three and minus
half is midway between 0 and minus 1 we
can also visualize the number system by
starting from the top that is from the
real numbers let me pull up the concept
board for you real numbers can be
divided into rational and irrational
numbers rational numbers can be divided
into integers and non integers now
integers can be divided into negative
integers and whole numbers whole numbers
can be split into zero and positive
integers which are natural numbers
coming back to the non integers
fractions and decimals these can be
divided into terminating decimals and
non terminating or recurring decimals
now let's say you want to find rational
numbers between any two given rational
numbers for example between 3 & 5 one
simple answer is 4 we can get it by
finding the average 3 plus 5 by 2 which
is equal to 4 if you want to find more
rational numbers then you can do 3 plus
4 by 2 that's 3 point 5 & 4 plus 5 by 2
which is 4 point 5 now let me pull up
some more interesting examples for you
if you want to find two rational numbers
between two fractions for example 1 by 7
& 4 bytes
since the denominators here are equal
and the numerators have a gap we can
fill in the fractions in the gap so our
answer is going to be 2 by 7 and 3 by 7
but what if there's no gap for example
if we have to find two rational numbers
between 1 by 7 and 2 by 7 then what do
you do we can create a gap by
multiplying the numerator and
denominator with a number since we need
two rational numbers we will multiply by
2 plus 1 which is 3 multiplying the
numerator and denominator of the two
fractions by 3 we get 3 by 21 and 6 by
21 again the denominators are the same
but can you see that we've got a gap in
the numerator so the two fractions we
can insert our 4 by 21 and 5 by 21 now
let's look at the case where the
denominators of the two fractions are
not the same for example if we want to
insert six rational numbers between 1/2
and 2/3 as you can see the denominators
are not equal so the first step is to
make the denominators equal so our two
fractions become 3 by 6 and 4 by 6
similar to our previous case now the two
denominators are equal but there's no
gap in the numerators so what should we
do that's right we need to multiply by a
number and since we need to insert six
rational numbers our number is going to
be 6 plus 1 which is 7 so let's multiply
the numerator and denominator of the two
fractions by 7 so here we have 21 by 42
and 28 by 42 so here are the six
rational numbers that we can insert
between these two fractions and on
simplifying the six rational numbers
and that's our answer let's talk about
fractions and decimals how to convert
fractions to decimals and decimals to
fractions converting fractions to
decimals is easy you just need to divide
for example 3 by 4 is 0.75 a terminating
decimal number 1 by 3 is 0.33 3
recurring a recurring decimal number now
if you want to convert 0.75 to a
fraction then we can write it as 75 by
100 simplifying we get 3 by 4 so these
are simple but a more interesting
question is how do you convert a
recurring decimal number to a fraction
for example 0.333 recurring now we know
that 0.3 is 3 by 10 0.33 is 33 by 100
0.333 is 333 by thousand but we are
looking for zero point 3 3 3 3 recurring
so let me show you the technique how to
convert a recurring decimal number to a
fraction coming up for you right now let
X be the recurring decimal number which
is 0.3 recurring in this case so let's
write it as X equal to 0.33 3 3 and so
on now we need another number where the
part after the decimal is the same so
let's multiply X by 10 and we have 10 X
equal to 3 point 3 3 3 and so on so as
you can see here X and 10x have the same
part after the decimal now let's
subtract X from 10x so I'm going to copy
the first line here
on subtracting we get 9x equal to 3 so X
is 3 by 9 and if we simplify we get 1 by
3 so we have converted our recurring
decimal number 0.3 recurring to a
fraction 1 by 3 let's look at another
example again let X be the recurring
decimal number which is one point two
seven recurring to find another number
with the decimal part matching this time
we can't multiply by 10 we need to
multiply by 100 and as you can see in
hundred X the decimal part matches so
once again subtracting X from 100 X we
get 99 X equal to 126 and on simplifying
we get 14 by 11 so we've converted our
recurring decimal number to a fraction
now that we are done with the topic of
rational numbers let's take a look at
the top three questions on this topic
coming up for you right now
I would like you to pause the video
right here and try solving these
questions let's make this more
interactive than just watching the video
so do post your answers in the comments
below or if you have any doubts
questions feel free to write it in the
comments below and I'm going to make a
commitment to answer all of them
promptly so I'm going to move off and
let you solve these questions thanks for
watching the whole video I hope the
number system is really easy for you now
and the numbers seem more rational and
natural to you all puns intended in that
sentence and do remember to Like comment
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