Question

Part A: Find a rational number that is between 9.5 and 9.7. Explain why it is rational.

Part B: Find an irrational number that is between 9.5 and 9.7. Explain why it is irrational. Include the decimal approximation of the irrational number to the nearest hundredth.

Answer 1

@jim_thompson5910

Answer 2

if a and b are rational numbers then look at (a+b)/2

Answer 3

Part B) To find an irrational number between x and y, find the geometric mean M

\[\Large M = \sqrt{x*y}\]

M is irrational if and only if x or y are not perfect squares

Answer 4

I think you mean M is irrational if and only if x*y is not perfect square

Answer 5

No I meant it like this

\[\Large M = \sqrt{x*y}\]
\[\Large M = \sqrt{x}*\sqrt{y}\]

In the second equation, sqrt(x) is only rational if and only if x is a perfect square. Same for y.

Answer 6

so if x wasn't a perfect square, or y wasn't, then M would be irrational

Answer 7

x=12, y=75
x*75=900=30^2

Answer 8

oh true. I didn't think of that. So yeah x*y not being a perfect square is a better condition

Answer 9

you could also use a known irrational...\(\pi= 3.141592...\)

then \[\pi/100=.03141592...\]

then \[9.5+\pi/100=9.53141592...\] which is irrational (it has a non-terminating non-repeating decimal expansion)