Video summary
The video introduces the concept of real numbers, which encompass both rational and irrational numbers, focusing specifically on their decimal expansions. It establishes a fundamental distinction between these two categories based on the nature of their decimals: a rational number always has a terminating or non-terminating recurring decimal expansion, whereas an irrational number possesses a non-terminating, non-recurring decimal expansion that continues infinitely without repeating any pattern. To illustrate this difference, the session uses specific examples such as 5/8, which converts to the terminating decimal 0.625 and is therefore rational, in contrast to root two ($\sqrt{2}$), whose value begins with 1.414 but never settles into a repeating cycle, confirming its status as an irrational number.
A significant portion of the discussion addresses how to identify infinitely many irrational numbers between any two given rational values by constructing non-recurring decimals within that specific range. Using the fractions 5/7 and 9/11 as boundaries, which correspond approximately to 0.7714... and 0.8181..., respectively, the presenter demonstrates a method for generating irrational numbers strictly between them. By creating decimal sequences like 0.75075075... or 0.76707670..., where digits are altered slightly after an initial repeating pattern to break the recurrence, one can generate valid irrational numbers such as 0.8084*0 that fall precisely within the interval defined by the two rational endpoints.
The final segment of the session explains the geometric representation of real numbers on a number line through a process of successive magnification. This technique involves starting with a broad range, such as between integers four and five for the number 4.2626..., and progressively dividing each interval into ten equal parts to narrow down the location of the target point. The presenter visualizes this by first isolating the region between 4.2 and 4.3, then further subdividing it to find the spot between 4.26 and 4.27, continuing until reaching the specific decimal places required; ultimately, through repeated subdivision and visualization of smaller intervals like those between 4.262 and 4.263, the exact point representing a complex real number is located on the line.
Read the full video transcript
hello and welcome to the session in this
session we will discuss real numbers and
their decimal
expansions we know that all the rational
and irrational numbers make up the
collection of real
numbers now we have decimal
expansion
of a rational number
is either
terminating
or
nonterminating
recuring moreover we can also say that a
number whose decimal expansion is
terminated or non-terminating recurring
is a rational number then decimal
expansion
of an irrational
number
is nonterminating
non
recing we can also say that a number
whose decimal expansion is
non-terminating non- recing is
irrational like consider the number 5
upon 8 now this is equal to
0.625 that is this is the decimal
expansion of this number now this
decimal expansion is
terminating so hence we can say that
this number 5 upon 8 is a rational
number now consider the irrational
number <
tk2 its decimal expansion
is given
by
1.414 21 1 3 5 6 2 3 7 and so on so as
you can see that the decimal expansion
of run2 is non-terminating non-
recurring we can find many irrational
numbers between two given numbers
suppose the two given numbers are 5 upon
7 and 9 upon 11 let's try and find out
three
different irrational
numbers
between 5 upon 7 and 9 upon
11 now 5 upon
7 has a decimal expansion 0.7 714 28 5
71
4285 and so on then 9 upon 11 has deim
expansion as
0.81
8181 81 82 and so on now to find
irrational number between these two
numbers we find a number which is
non-terminating non- recording line
between these two
numbers we can find infinitely many such
numbers like the first irrational number
between these two numbers given by
0.75 0
75 0 75 Tri 0 and so on then the second
irrational number can be given by
0.76 6 7 0
767 0
767 Tri 0 and so on then the third
irrational number is given
by 0 8 0 8 0 8 0 8 4 * 0 and so
on so this is how we find the irrational
numbers between given two
numbers now we discuss representing real
numbers on the number
line we know that every real
number is
represented by a unique point
on the number
line
further every
point on the number
line
represents one and only one
real
number consider the number
4.26 bar we need to visualize this
number on the number line up to four
decimal places so up to four decimal
places this number can be written as
42626 now you can clearly see that this
number would lie between the numbers
four and
five that is this number lies in the
this area that is between the numbers
four and
five now we draw another number line to
visualize the area between the numbers
four and five
now we divide this region between the
numbers four and five into 10 equal
parts so this is the middle point of
this number
line which represents the number
4.5 now give number is 4.26 26
so thus we can say that this number
would lie between the numbers 4.2 and
4.3 that is this region now we will draw
another number line which represents the
region between 4.2 and 4.3 and we
visualize this
region again we will divide this region
into 10 equal parts now this is the
middle point 4 point
25 this is the middle point of this
number
line now we have reached the location
4.26 now
4262 would lie in the Region
4 26 to 4.27 that is this
region now we draw another number line
to visualize the region between the
numbers 4.26 and
4.27 now the Middle Point
of this number line is given by the
number
4.26
5 now we have reached the number
4262 now
42626 would lie in this
region that is between the numbers 4.
262 and
4263 now we will visualize the region
between the numbers 4.26 2 and 4.26 3
to get the required number on the number
line now this is the middle point of
this number line representing the number
42625 this is the required point which
represents the number
42626 so as you can see we started from
this number
line where we magnified or visualize the
region between the numbers four and five
then we visualize the region between the
numbers 4.2 and
4.3 then we visualize the region between
the numbers 4.26 and
4.27 and after that we visualize the
region between the numbers 4.26 2 and
4.26 3 and from there we get the
required point 4.26
26 this completes the session hope you
have understood real numbers and the
decimal expansions and how we represent
real numbers on the number line