Question

How to compare real numbers?

Answer 1

Hint: Here we have to define the real number and discuss some examples to compare whether it is a real number or not. After discussing the example for each and every term, we will conclude the required answer for comparing the real numbers.

Complete step-by-step solution:
Real numbers are numbers which include rational and irrational numbers both.
Rational numbers include integers $\left( { - 3, - 2} \right)$, whole numbers$\left( {3,4,5} \right)$, fractions and decimals.
Irrational numbers include numbers like$\sqrt 2 ,\sqrt 3 ,{\text{etc}}$.
Let us take two fractions as examples to compare. The two fractions are: $\dfrac{5}{6}$ and $\dfrac{8}{5}$,
Now we can only do comparisons between two numbers when they both have the same denominator thus we need to have a common denominator for both the fractions.
In order to get a common denominator, we find the L.C.M of the denominators $6,5$
$
  5\left| \!{\underline {\,
  {5,6} \,}} \right. \\
  6\left| \!{\underline {\,
  {1,6} \,}} \right. \\
  1,1 \\
 $
On further multiplying, we get$5 \times 6 = {\text{30}}$, thus the common denominator is 30
Thus, in order to put this common denominator in our respective fractions, we also need to multiply the numerator with the same number as the denominator.
Thus, $\dfrac{{5 \times 5}}{{6 \times 5}} = \dfrac{{25}}{{30}}$ and $\dfrac{{8 \times 6}}{{5 \times 6}} = \dfrac{{48}}{{30}}$
Comparing two whole numbers is simple but in order to compare fractions, we need to have a common denominator.
Now that both the fractions have the same denominator, we can compare them easily:
$25 < 48$
$\therefore \dfrac{{25}}{{30}} < \dfrac{{48}}{{30}}$
Thus, $\dfrac{5}{6} < \dfrac{8}{5}$
Let us take another example with two decimals: $4.5{\text{ and }}3.980$
In order to compare these two decimals, let us first ensure if they have the same amount of numbers after the decimal point. $4.5$ has one number after the decimal point while $3.980$ has three numbers after the decimal point.
Thus we add two more zeroes after the decimal point in $4.5$⇒$4.500$
Now, we can easily compare both the decimal numbers.
Thus, $4.500 > 3.980$
Now, in order to compare irrational numbers, let us take two numbers as an example:$\sqrt {2{\kern 1pt} } $ and $\sqrt 3 $
 We know that the value of $\sqrt {2{\kern 1pt} } $= $1.414$ and value of $\sqrt 3 = 1.732$
Comparing both the values, we find $1.414 < 1.732$
Thus, $\sqrt 2 < \sqrt 3 $
Hence we conclude the comparison of real numbers.

Note: In order to compare irrational numbers, it is important to remember the values of the common irrational numbers like $\sqrt 2 ,\sqrt 3 $ etc.
Real numbers can be compared easily if we know what kind of a number we are dealing with.
In order to compare decimal numbers; we should make sure that we have the same amount of numbers after the decimal point.