Is the square root of 945 irrational?

In summary: Therefore it cannot be exactly equal to 945, so sqrt(945) is irrational.In summary, the square root of 945 is irrational, as confirmed by the fact that it cannot be represented as a fraction and the fundamental theorem of arithmetic. While a calculator may approximate it as a fraction, it is not an exact representation due to the limitations of significant digits.
  • #1
srfriggen
306
5
Is the square root of 945 irrational?

I feel it is rational because my TI-84 Plus converts it into 275561/8964, however, I am unsure whether the calculator is estimating.

Can someone please advise. It can be broken down into 3√105, and again, my calculator is able to convert √105 into a fraction.

Thank you.
 
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  • #2
Yes, sqrt(945) is irrational. Your calculator is finding a fraction approximating sqrt(945).

sqrt(945) = 30.740852297878796...
275561/8964 = 30.740852298081215...

They are not equal.
 
  • #3
If the square root of an integer is rational it must be an integer. The square of this fraction is 945.000000012445056. You are running into the limits on the calculator significant digits.
 
  • #4
I typed these 3 commands into worlfram
https://www.wolframalpha.com/input/?i=prime+factorization+of+8964
https://www.wolframalpha.com/input/?i=prime+factorization+of+275561
https://www.wolframalpha.com/input/?i=prime+factorization+of+945

so supposing it is rational we get

##3^3 \cdot 5 \cdot 7 = 945 = \big(\frac{275561}{8964}\big)^2= \big( \frac{11\cdot 13\cdot 41 \cdot 47}{2^2 \cdot 3^3 \cdot 83 }\big)^2##

clearing the denominator gives

##\big(3^3 \cdot 5 \cdot 7\big)\big(2^2 \cdot 3^3 \cdot 83 \big)^2 = 945\big(2^2 \cdot 3^3 \cdot 83 \big)^2 = \big( 11\cdot 13\cdot 41 \cdot 47\big)^2 ##

but this violates the fundamental theorem of arithmetic and hence what your calculator gave cannot be an exact rational expression for the square root
 
  • #5
Two-line approach: 275561/8964 is a fraction that doesn't share prime factors in numerator and denominator. Therefore (275561/8964)2 is also a fraction with this property and the numerator cannot be a multiple of the denominator.
 

1. What is an irrational number?

An irrational number is a number that cannot be represented as a ratio of two integers, meaning it cannot be written as a fraction. These numbers have decimal representations that neither terminate (end) nor repeat.

2. How do you determine if a square root is irrational?

To determine if a square root is irrational, you need to calculate the decimal representation of the square root. If the decimal digits do not terminate or repeat, then the square root is irrational.

3. Is the square root of 945 a rational or irrational number?

The square root of 945 is an irrational number because its decimal representation does not terminate or repeat. It is approximately 30.748, and the decimal digits continue infinitely without any pattern.

4. Can you simplify the square root of 945?

No, the square root of 945 cannot be simplified further because it is an irrational number. It cannot be written as a fraction or a whole number.

5. Why is it important to know if a number is rational or irrational?

Knowing if a number is rational or irrational is important in many fields of science and mathematics. It helps in solving equations, understanding patterns in numbers, and making accurate calculations. It also has applications in fields such as engineering, physics, and computer science.

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