Question

How do you know if a square root is rational or irrational?

Answer 1

Use the definition of fraction.

If a real number cannot be represented by a fraction, then it is irrational. Otherwise, it is rational.

Answer 2

Do you have a calculator?

Answer 3

yes i do

Answer 4

however \[\sqrt2=\frac{\sqrt 2}{1}\] is irrational

Answer 5

So is the number 0.6767 rational or irrational?

Answer 6

rational...
0.6767 = 6767/10000

Answer 7

\[0.6767=\frac{6767}{1000}\]

Answer 8

the number is irrational if doesn't have a decimal expansion that repeats after a while

Answer 9

I think it's 10000 for the denominator.

Answer 10

yeah your right

Answer 11

1000 not 10000

Answer 12

6767/1000 = 6.767 , isn't it?

Answer 13

\[\frac{6767}{1000}=6.767\]

Answer 14

Anyway, the main point is that 0.6767 can be expressed in fraction form

Answer 15

But is it rational or irrational? I am confused

Answer 16

rational

Answer 17

some numbers like one seventh
might look irrational from their decimal expansion
but they will eventually repeat.
\[\frac 17=0.1428571428571428471.....=0.\dot14285\dot7 \]
while the square root of two
\[\sqrt 2\approx1.414213562373095048801688724209\]
will never repeat , because it is an irrational number

Answer 18

\[\pi\approx3.14159\]\[e\approx2.71828\]\[\phi\approx1.61803\]are some example of irrational numbers that are not surds

Answer 19

Thank you everyone for your amazing explanations and answers!! :D