# System of numbers problem

#### HazeTheBeater

##### New member
Hello everyone Currently im studying the system of numbers. In my workbook of McGrawHill that I bought, theres this last problem thats asking me if the square root of 10 is a natural number. The result of this root is 3.16 which isn't a natural number but a rational number instead? I don't understand since Natural numbers only rang from 1 to plus infinity.

Thanks a lot, I know its more of a theory question, but I would gladly accept some help

#### MarkFL

##### Super Moderator
Staff member
If the square root of 10 is a natural number, then you should be able to find some natural number whose square is 10. Otherwise it will be an irrational number. I would observe that:

$$\displaystyle 3^2=9$$

$$\displaystyle 4^2=16$$

And so there is no natural number whose square is 10.

#### JeffM

##### Elite Member
Hello everyone Currently im studying the system of numbers. In my workbook of McGrawHill that I bought, theres this last problem thats asking me if the square root of 10 is a natural number. The result of this root is 3.16 which isn't a natural number but a rational number instead? I don't understand since Natural numbers only rang from 1 to plus infinity.

Thanks a lot, I know its more of a theory question, but I would gladly accept some help
I think you have some misunderstanding about different number systems.

The basic one is the system of natural numbers, which is defined either as all whole numbers greater than zero or as all whole numbers greater than or equal to zero. (The first definition is more common, but perhaps slightly less useful.)

Depending on your book, the next system usually defined is either the integers or the non-negative rational numbers. But one of the great early discoveries of Greek mathematics is that the roots of most natural numbers are NOT rational numbers, but something called irrational numbers. In other words, the so-called real numbers consist of integers, the rational numbers that are not integers, and the irrational numbers, none of which are integers or rational numbers.

If you know the meaning of the phrase "proper subset":

Natural numbers are a proper subset of the integers $$\displaystyle \mathbb N \subset \mathbb Z$$

Integers are a proper subset of the rational numbers $$\displaystyle \mathbb Z \subset \mathbb Q$$

Rational numbers are a proper subset of the real numbers $$\displaystyle \mathbb Q \subset \mathbb R$$

#### Mr. Bland

##### Junior Member
The [square root of 10] is 3.16 which isn't a natural number but a rational number instead?
A natural number is some whole number (not a fraction) that is not negative and, depending on who you ask, may include zero. The square root of 10 is positive, but it falls between 3 and 4. It is not a whole number, and therefore is not a natural number.

The square root of 10 has more digits than just those three. It's actually 3.16227766016837933... well, I can't type them all out here because they just keep going forever and ever, and they ultimately do not repeat.

Any whole number, including the natural numbers and negative whole numbers, is known as an integer.

A number that can be expressed as a division of integers is called a rational number (which includes the integers and all fractions). Sometimes these make short and simple figures in decimal form, but sometimes they go on forever. However, if a fraction does go on forever in decimal form, then the sequence of digits is guaranteed to repeat after a certain point.

Otherwise, if a number is not a whole number and the digits never repeat, that's what's called an irrational number. It happens to be that, for any natural number that is not itself the square of a natural number, its square root will always be irrational.

#### lookagain

##### Elite Member
A natural number is some whole number (not a fraction) ...
All whole numbers are expressible as fractions.

#### Mr. Bland

##### Junior Member
But not all fractions are whole numbers. Unless the argument is that whole numbers should be introduced to people as "any nonnegative rational number in the form $$\displaystyle \frac{x}{1}$$", there's a meaningful distinction between whole numbers and fractions.

#### lookagain

##### Elite Member
But not all fractions are whole numbers.

The set $$\displaystyle \ \ \{0. \overline{9}, \ \tfrac{2}{2}, 1\} \ \$$ contains one element. It is the whole number one.