Rational vs. Irrational - Decimals in Square Roots?

[teacherk2]

Hello,

Apologies if this is not in the right forum. Asking so I, as a paraeducator, can make sure I'm explaining correctly to my students (7th graders, language learners), as the math classes I service are very large with a teacher new to this content area.

If solving a square root results in a decimal, is that always irrational, even if it's a short and manageable decimal? I've been trying to explain to my students that an irrational number has any of the following 3 qualities:
1. Is, or contains, the numeral pi
2. Is a long, continuing decimal that does not repeat or have a pattern
3. Is a non-perfect square root (the example is give to explain a "non-perfect square root" is that the square root of 100 is 10 and that is an easy, simple number -- so that is a perfect square, but the square root of 200 gives us 14.142... which gets long and crazy)

I then emphasize that these are "irrational" numbers because they're big and long and without a pattern/repetition, we don't know where they'll end, which I know is not exactly the definition, but at this grade level works well for what we're using it for. (If this is still incorrect, please let me know! I, honestly, also learned of it this way and only have a loose grasp of the "without a ratio" definition that I've seen.)

However, the teacher today in the lesson mentioned that a square root that has any decimal would be irrational. I admittedly can't think of any square root they would be solving that would have a simple decimal, like x.25, but I was hoping to get some clarity on this. Is a decimal in a solved square root, even if it is simple, always irrational, or is only if it is a continuing decimal without a pattern / without repetition?

Thanks! I wish a resource like this would have existed (or, would have been easier to find) when I was in school.

 

[Cubist]

However, the teacher today in the lesson mentioned that a square root that has any decimal would be irrational. I admittedly can't think of any square root they would be solving that would have a simple decimal, like x.25, but I was hoping to get some clarity on this. Is a decimal in a solved square root, even if it is simple, always irrational, or is only if it is a continuing decimal without a pattern / without repetition?

Thanks! I wish a resource like this would have existed (or, would have been easier to find) when I was in school.

Great question!

I know that you're teaching at pre-algebra level but I'm going to show you a little bit of algebra to answer the question (I'm not recommending that you show algebra to your class). Perhaps you can just show your class the end result, without the algebra, as a counter example to the teacher's statement...

If I wish the result of a square root to be 0.7 then...
[math]\sqrt{x}=0.7\\ \sqrt{x}=\frac{7}{10}\\ \text{square both sides giving }x=\left(\frac{7}{10}\right)^2\\ x=\frac{7^2}{10^2}\\ x=\frac{49}{100}\\ x=0.49[/math]
You can type that value for x into a calculator to verify (type 0.49 then press "sqrt" button). Sometimes a calculator might have rounding issues (because of how it works) and it might show something silly like 0.7000000001 however the exact answer would still be 0.7. You could try this with different decimals until you find one that displays properly on the calculators that you might use in class.

EDIT: To do this quickly on a calculator, you can just type 0.7 then press [imath]x^2[/imath] button the obtain 0.49 which will then turn back into 0.7 with the [imath]\sqrt{x}[/imath] button

 

[JeffM]

This is a highly personal reply.

Irrational numbers do not correspond to anything visible in the physical universe. They arise as necessary to describe the ideal world of mathematics.

Every rational number can be exactly represented as a fraction with a whole number in the numerator and a whole number not equal to zero in the denominator. An irrational number cannot be exactly represented that way. In short, the rational numbers are the set of proper and improper fractions.

In terms of decimal representation, rational numbers either terminate or have an infinitely repeating finite pattern. For example,
1/3 = 0.33333 forever. The decimal representations of irrational numbers neither terminate nor have any finite pattern nor do they correspond to any fraction involving whole numbers. Consequently, when we do arithmetic with irrational numbers, we have to use approximations.

There are more irrational numbers than rational numbers, but the only irrational numbers that are even remotely relevant to children of the indicated age are

[imath]\pi \approx 3.1416[/imath] and the square and cube roots of the whole numbers that are not perfect squares or cubes.

I am not sure what, if any, is the the pedagogic value of introducing the concept of irrational numbers to children of this age. But we certainly want to keep it simple.

Rational numbers are proper or improper fractions of whole numbers (excluding division by 0.)
Irrational numbers are not fractions of whole numbers.

 

[Steven G]

1. Is, or contains, the numeral pi
2. Is a long, continuing decimal that does not repeat or have a pattern
3. Is a non-perfect square root (the example is give to explain a "non-perfect square root" is that the square root of 100 is 10 and that is an easy, simple number -- so that is a perfect square, but the square root of 200 gives us 14.142... which gets long and crazy)

1) Not true. 3pi/pi clearly contains pi but it is not irrational since 3pi/pi = 3
2) Not true. In my opinion, a decimal that ends after 20 digits is long. Others may say that a 100 digit decimal is long...
Long is not well defined at all. You mean to say, and should say, that a non-ending decimal number that does not repeat is irrational.
.1010010001000.... does have a pattern and it is irrational. So please remove pattern from your definition of irrational.
3) No long and crazy. You need to replace long with non-ending. What you call an easy simple number is called an integer.

You are trained as an educator and you probably are very good at that.
My issue is that an educator of say mathematics does not have to take enough math courses to even get down mathematical terminology correctly.
I think that educators should also do a degree in the discipline that they are teaching.
Again, this is not about you but rather the requirements to teach.

 

[Steven G]

This is a highly personal reply.

Irrational numbers do not correspond to anything visible in the physical universe. They arise as necessary to describe the ideal world of mathematics.

Every rational number can be exactly represented as a fraction with a whole number in the numerator and a whole number not equal to zero in the denominator. An irrational number cannot be exactly represented that way. In short, the rational numbers are the set of proper and improper fractions.

In terms of decimal representation, rational numbers either terminate or have an infinitely repeating finite pattern. For example,
1/3 = 0.33333 forever. The decimal representations of irrational numbers neither terminate nor have any finite pattern nor do they correspond to any fraction involving whole numbers. Consequently, when we do arithmetic with irrational numbers, we have to use approximations.

There are more irrational numbers than rational numbers, but the only irrational numbers that are even remotely relevant to children of the indicated age are

[math]\pi \approx 3.141592654[/math]
and the square and cube roots of the whole numbers that are not perfect squares or cubes.

Irrational numbers do not correspond to anything visible in the physical universe.
Really? I will try to be gentle! Have you ever seen a right triangle whose legs are both the same length in the universe we live in? ....

 

[JeffM]

Irrational numbers do not correspond to anything visible in the physical universe.
Really? I will try to be gentle! Have you ever seen a right triangle whose legs are both the same length in the universe we live in? ....

No, and neither have you. Every measurement has some degree of uncertainty. How do you even begin to determine whether the legs of a purported triangle are straight lines. You cannot use rays of light because they are bent by gravity.

 

[Steven G]

No, and neither have you. Every measurement has some degree of uncertainty. How do you even begin to determine whether the legs of a purported triangle are straight lines. You cannot use rays of light because they are bent by gravity.

You can construct such a triangle with a compass, straight edge and paper. At least this what my geometry teacher told me. And you know me, I believe everything my high school teachers told me.

 

[Steven G]

If solving a square root results in a decimal, is that always irrational, even if it's a short and manageable decimal?
As already pointed out, the answer to your question is no.

(1/2)*(1/2) = 1/4 = .25. We can then conclude by the way that I got .25 that sqrt(.25) = 1/2 or .5
Again, short and manageable are not mathematical terms. Being a short decimal number is relative.
.127329909154317951*.127329909154317951=0.01621290576524686233960694800083840100. So sqrt(0.01621290576524686233960694800083840100)=.127329909154317951
I would say that 0.01621290576524686233960694800083840100 is NOT a short decimal number, yet the sqrt of this number is rational.

 

[JeffM]

You can construct such a triangle with a compass, straight edge and paper. At least this what my geometry teacher told me. And you know me, I believe everything my high school teachers told me.

LOL

First, you need to find a perfectly smooth piece of paper, a pencil that draws a line of zero breadth, a compass that is impervious to pressure, and a perfectly straight straight-edge, one that is has not a single atom out of true.

Those are supplied to high school teachers, but not to anyone else.

?

 

[Deleted member 4993]

LOL

First, you need to find a perfectly smooth piece of paper, a pencil that draws a line of zero breadth, a compass that is impervious to pressure, and a perfectly straight straight-edge, one that is has not a single atom out of true.

Those are supplied to high school teachers, but not to anyone else.

?

Moreover the space is curved - the nature of straight-line will change!

 

[Cubist]

@teacherk2 is this topic on a syllabus that you must adhere to? Or, is this topic flexible and your goal is to give the children a glimpse into the subject of mathematics? If the latter, then perhaps some people here might be able to suggest some fun maths activities/ topics.

 

[Otis]

a square root results in a decimal, is that always irrational…?

Hi TK2. The short answer is 'no'.

As your students are language learners, I think it's important to focus on the meaning of words and names.

All numbers* (Rational or Irrational) have decimal forms, with decimal-forms of Rational numbers needed (as approximations) to express Irrational numbers in a decimal form. It seems like you're misusing the phrase "a decimal" above to mean "a number that contains at least one non-zero digit to the right of the decimal point". In other words, a mixed number (see below).

I like the distinction Jeff made, by using the name 'Whole numbers'. Here is the set of Whole numbers:

{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, …} with the ellipsis dots indicating that the pattern continues with no end.

Every number* (including Whole numbers) may be written in a decimal form, but again, decimal forms of Irrational numbers are only approximations. They are not exact.

For examples, when I enter √16 on a calculator, the result is displayed as 4.0 (a decimal form of the Whole number 4). Hence, just because we see a decimal point (on a calculator or otherwise) does not mean that the number containing it is Irrational. Also, Pi is an Irrational number. When someone says, "Pi equals 3.1429", it probably means "for me, the Rational number 31429/10000 in decimal form is a good decimal approximation for the Irrational number Pi".

Regarding mixed numbers: All Rational numbers strictly in-between 0 and 1 are not whole. They are proper fractions. When we add a proper fraction to a Whole number, we get what's called a mixed number (i.e., a "mixture" of a Whole number and a fractional part), like 8.5, which is [imath]8\frac{1}{2}[/imath].

All numbers* in-between two consecutive Whole numbers are either mixed numbers (which are Rational) or Irrational numbers.

Have your students already learned about how we categorize numbers at their level? Can they define the set of Real numbers in their own words? That is, are they familiar with the subset names: Natural numbers, Whole numbers, Rational numbers. As the linked reference shows, all Natural numbers are also Whole and Rational, and all Whole numbers are also Rational. Sets within sets. Again, any number in those subsets may be expressed exactly with a decimal point.

If your students understand these categorizations of numbers, then the definition of an Irrational number (at their level) is simply this: Any number that is not Rational (as Jeff posted).

And, Irrational numbers cannot be expressed exactly using a decimal point.

If they have not yet learned the meaning of such language, then I think teaching those names and related math words first is tantamount. Introducing the concept of the Real number line may also be useful, in your discussions, as it provides a graphical perspective (visual aid).

Cheers

*When I say, "all numbers" or "every number", I specifically mean Real numbers (the Rationals and the Irrationals), which excludes special numbers not taught at the Pre-Algebra level (eg: Imaginary numbers, vectors, etc.)
[imath]\;[/imath]

 

[Dr.Peterson]

However, the teacher today in the lesson mentioned that a square root that has any decimal would be irrational.

I've been trying to explain to my students that an irrational number has any of the following 3 qualities:
1. Is, or contains, the numeral pi
2. Is a long, continuing decimal that does not repeat or have a pattern
3. Is a non-perfect square root (the example is give to explain a "non-perfect square root" is that the square root of 100 is 10 and that is an easy, simple number -- so that is a perfect square, but the square root of 200 gives us 14.142... which gets long and crazy)

When irrational numbers are first introduced, it is (unfortunately) common to provide a few examples, namely, pi and square roots OF INTEGERS that are not whole numbers. They are then used so much that the impression is given that these are the only examples, and that any non-whole square root is irrational. This is wrong.

The reality is that MOST numbers on the real number line are irrational. It should be taught clearly that the examples given are only examples chosen because they are easier to talk about than most; and that in general, proving that a number is irrational is quite difficult. The students should be aware primarily only that irrational numbers exist, and what the definition is. And I'm not sure how useful even that knowledge is.

 

[JeffM]

I like all the responses.

Here is the issue. People started thinking about mathematics as someting more than just arithmetic and formulas sometime around five thousand years ago. These were bright people. It took them perhaps 2500 years to conceive of numbers that could not be whole number multiples of some other whole number. There is a report that this idea was so shocking that the first mathematician to demonstrate its necessity in the ideal world of mathematics was executed for impiety.

A concept that resisted recognition by trained adult minds for 2500 years is not going to be easy to explain to only partially trained 12 year olds. I truly believe (with Kronecker) that the only numbers that can be empirically validated are the whole numbers and that the rest of the menagerie of types of numbers “exist” only in the idealized world of mathematics. But these numbers have turned out to be of immense practical importance because they let us escape any obsession with actual irrelevancies that have no material significance.

 

[Cubist]

Every number* (including Whole numbers) may be written in decimal form, with any digits appearing to the right of the decimal point as zeros.

I agree with your post except for the line above. Surely irrational numbers can only be approximated in decimal form EDIT: I suspect that you were talking about every whole number and just made a simple typo.

 

[JeffM]

This unfortunate teacher has been plunged into very deep ontological and epistemological waters. Personal opinion. We should explain the progression from whole numbers to rational to algebraic numbers as necessary to give answers to certain kinds of problems. Then moving on to irrational, complex, quaternions, etc. is simply an extension to getting answers to ever more complex problems.

 

[Otis]

I agree with your post except … irrational numbers can only be approximated

I agree with you completely! Thank you.

In my mind at the time, 'decimal form' included approximations. I see now that I ought to have been specific. (I'll get that fixed.)


[imath]\;[/imath]