DoubtProve the term is irrational

In summary, the conversation discusses how to prove that a number with zeroes only in decimal places numbered 10^n is irrational. The participants suggest that because the number of zeroes does not follow a definite pattern, it cannot be represented as a rational number. One participant also mentions that all rational numbers have periodic expansions, but since this number does not follow a pattern, it must be irrational. Another participant suggests that if the number were rational, it would have a repeating period, but this is not possible because there are more digits between 10^n and 10^(n+1) than the period.
  • #1
vio
7
0

Homework Statement


Prove that any number with zeroes standing in all decimal places numbered 10^n and only in these places is irrational?(yeah,its the easiet question in my list,but I am still not sure about it)


Homework Equations





The Attempt at a Solution


when i think about it,since the number of zeroes don't follow a definite pattern,i mean the same pattern,it will be difficult to represent it as a rational,since one never knows the where the next zero is ,or how its repeatin??..since it not periodic,it must be irrational..
Well,,when i said that to myself,it seems like i waz just reading out sumthn from a text,i still didnt understand it well enough or sumthin..i know its very basic,but can someone explain it in simpler language?
 
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  • #2
u haz allreddy said everythin it cnt be peridic, cn it? all rationals haz peridcic expanshions. Maybe I forgot to misspell something there, sorry.
 
  • #3
Dick said:
u haz allreddy said everythin it cnt be peridic, cn it? all rationals haz peridcic expanshions. Maybe I forgot to misspell something there, sorry.
Fantastical reply, Dick !
 
  • #4
thanx!

Ohk...Thanx guyz... :)
(still somthings buggin me though )
 
  • #5
vio said:
Ohk...Thanx guyz... :)
(still somthings buggin me though )

Maybe just spelling out why it can't be periodic? If it's rational and it contains a zero digit that zero digit will repeat at some period p once you get into the repeating part. Pick n so that 10^n>p. Then the digits between 10^n and 10^(n+1) will have no zero digits. But there's more than p of them. So it can't be periodic.
 
  • #6
SammyS said:
Fantastical reply, Dick !

Thanks. It was hard to resist...
 

Related to DoubtProve the term is irrational

1. What does it mean for a number to be irrational?

For a number to be irrational, it means that it cannot be expressed as a ratio of two integers (whole numbers). In other words, it cannot be written as a fraction with a finite number of digits in the numerator and denominator. Irrational numbers have infinite decimal representations that do not repeat or terminate.

2. How do you prove that a number is irrational?

To prove that a number is irrational, you must show that it cannot be expressed as a rational number. This can be done through various methods such as the proof by contradiction, which assumes that the number is rational and then shows that this leads to a contradiction. Other methods include using the properties of irrational numbers, such as the fact that they are non-repeating and non-terminating.

3. Is pi an irrational number?

Yes, pi (π) is an irrational number. It has been proven that pi cannot be expressed as a ratio of two integers and it has an infinite decimal representation that does not repeat or terminate. This makes it one of the most well-known irrational numbers.

4. Can a number be both rational and irrational?

No, a number cannot be both rational and irrational. A number is either rational or irrational, there is no overlap between the two. Rational numbers have finite decimal representations or repeating decimal patterns, while irrational numbers have infinite non-repeating decimal representations.

5. How do irrational numbers affect mathematics and science?

Irrational numbers play a crucial role in mathematics and science. They allow us to accurately measure and calculate various quantities that cannot be expressed as whole numbers or fractions. For example, pi is essential in geometry and trigonometry, while the square root of 2 is used in the Pythagorean theorem. In science, irrational numbers are used in various calculations, such as in physics and engineering.

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