True or False: √17 ϵ Real numbers *?

Answer 1

The answer is true.

The reason is because sqrt(17) is an irrational number, and real numbers by definition are the set of rational and irrational numbers.

Answer 2

I am assuming that square in your question is an "is an element of" sign.

The equation x^2 - 17 = 0 has two roots, and they are distinct because the derivative 2x = 0 has 0 for a root, which does not satisfy the original equation. But algebraically there is no way to tell them apart. As long as you did only arithmetic operations and not comparisons, you could use -√17 as the square root of 17 that is close to 4.1.

One uses limits or a similar concept to identify a positive real number whose square is 17. One way is to define real numbers as cuts of rational numbers - a partition of the reals into two sets such that all elements of one are greater than all elements of the other. The sets {x|x<2| and {x|x>=2} form a cut that can be identified with the rational number 2. The sets {x|x>=0 and x^2<17} union {x|x<0} and {x|x >= 0 and x^2>=17} form a cut which does not correspond to any rational number. But if you square it (using special rules for multiplicaiton in cuts), you get 17. This number is positive, so you can take this number to be √17 . But remember that it is by taking limits or cuts on infinite sets that you can identify a real number whose square is 17. It does not come out algebraically.

So yes, √17 is a real number.

Answer 3

Yes, it's a real number. It's a real number on the number line between 4.1 and 4.2, even though it's not an integer and not rational.

Answer 4

true

Answer 5

true

Answer 6

T