Prove that if x and y are rational Numbers, then prove that (x+y)/2 is also a rational number between x and y?
Answers
Concept:
A rational number is a sort of real number that has the form p/q and q does not equal zero.
Given:
x and y are two rational numbers.
Prove:
\frac{(x+y)}{2}
2
(x+y)
is a rational number between x and y.
Solution:
If x and y are rational numbers, then x and y may be written in the form of \frac{p}{q}
q
p
where p and q ∈ Z.
Let x=ab and let y=cd where a,b,c,d ∈ Z.
Add x and y
x+y=\frac{a}{b} +\frac{c}{d}x+y=
b
a
+
d
c
x+y=\frac{ad+bc}{bd}x+y=
bd
ad+bc
ad+bc ∈ Z and bd ∈ Z.
∵ The multiplication and addition of two integers is again an integer.
Therefore x+y=\frac{a}{b} +\frac{c}{d}x+y=
b
a
+
d
c
is a rational number.
koCheck \frac{(x+y)}{2}
2
(x+y)
lies between x and y
Assume x=2x=2 and y=4y=4
\frac{(x+y)}{2}=\frac{4+2}{2}
2
(x+y)
=
2
4+2
\frac{(x+y)}{2}=\frac{6}{2} =2
(x+y)
=
2
6
=3
which lies between 2 and 4.
Hence if x and y are two rational numbers then \frac{(x+y)}{2}
2
(x+y)
is a rational number between x and y.