At our roots, aren't we all a little irrational?
So R>0, the Non-Negative Real number set, is joined by, R, the Real number set; N, Natural numbers; Q, Rational numbers (R was taken); Z, Integers (don't ask); P, Primes; and C, Complex numbers.
I also saw a notation for H, denoting Quaternions -- I didn't even go there. Except to note that Q was taken already because R was taken already.
Come back often for more funny math and geeky comics.
Most whole numbers have square roots which are irrational numbers, but not everything with a radical is irrational. The square root of a perfect square is perfectly rational. So how can you tell if a number is a perfect square without a calculator?
One way is through prime factorization. (Remember those factor trees from a long time ago. C'mon, they were fun to do -- and you can do them again.... just not when you're typing in a blog. Then, they're kind of a pain, but I'll try.)
Take a number such as 60. It's prime factorization is 2 X 2 X 3 X 5, or 22 X 3 X 5.
If we were to square 60, we'd multiply 60 X 60, but we could also multiply 22 X 3 X 5 X 22 X 3 X 5.
That number (3600) would have a prime factorization of 24 X 32 X 52.
Notice what happened to the exponents. They've all doubled from 1 to 2 or 2 to 4. Every time you square a number, the exponents of its prime factors double. So if a number has been square, then all of the exponents of its prime factors will be even numbers because they are multiples of two.
Going back to our original number, is the square root of 60 a rational number?
It could only be a rational number if 60 were a perfect square, and it can only be a perfect square if all the exponents of its prime factors are even. However, the prime factorization is 22 X 3 X 5. Only one factor is even, so it is not a perfect square and the square root of 60 is irrational.
But wait! There's more!
As long as we've done the legwork, there is one more thing that we can do. Radicals that are irrational can be simplified. This is done by factoring out the largest perfect square. If we look back at the prime factorization, 22 X 3 X 5, we can see that there are two factors of 2.
So the square root of 60 is the same as (the square root of 22) X (the square root of 3 X 5). The square root of 22 is just 2.
That means that the square root of 60 is (2) times (the square root of 3 X 5), or 2(radical 15).
But what about numbers which are written as decimals?
Every middle schooler remembers the joys of converting fractions to decimals (and to percents, but that's a post for a different day). But what about converting them back to fractions?
Terminating decimals are a cinch. Take a number like .375, which is properly pronounced three hundred seventy five thousandths. (It's usually pronounced "point three seven five" by lazy people, including some educators, including myself. Shame on me.) They can easily be turned into a fraction where the denominator is a power of 10.
There are three decimal places, so if you multiply the decimal by 103, you have the numerator 375 and 1000 as the denominator: 375 / 1000, which simplifies to 3/8, which is obviously a rational number.
Repeating decimals are a little trickier. Sure, you may recognize 0.7777777... and know that it's 7/9, or even 0.36363636... is 4/11. But what a strange number like 0.378378378...?
There is a simple procedure for it, involving a little Algebra.
Let x = 0.378378378378... And then do the following calculations:
By multiplying and subtracting, the repeating pattern of the decimal can be removed, leaving a multiple of x that will always be one less than a power of 10. Dividing by the coefficient transforms the value into a fraction (which may or may not be reducible).
This can be done for any repeating decimal. So any repeating decimal can be written as a fraction, i.e., a ratio of two integers and is, therefore, a rational number.
Moving on to another Common Core Algebra standard brings me to N-RN.3, which reads
The key word here is "explain". Before we do that, let's establish that the rest of the standard is true: if you add two rational numbers, you will get an irrational number; the sum or product of a rational and irrational number is irrational. (Leaving out the obvious case of multiplying by zero.)
If we remember that the word "rational" comes from "ratio" and that ratios can be written as fractions with numerators and denominators that are integers. Then the sum of any two rational numbers can be written as the sum of two fractions. All we need to add these two fractions is a Common Denominator, which can be obtained by multiplying the denominators. The set of integers is closed under multiplication; the product will be another integer. If we add the numerators, the set of integers is also closed under addition. The total will be another ratio of two integers, which must be a rational number.
To show that the sum of a rational number and an irrational number is irrational, we can use contradiction. Supposed that r is rational and x is irrational. (Let's not use i as it has a different meaning, which may cause confusion.) Assume there is a sum r + x that is rational. If we add -r, which is also rational, we get r + x - r, or just x, which must be rational because the sum of two rational expressions is rational. This is a contradiction because we started with x as irrational. So the sum of a rational number and an irrational number must be irrational.
The same contradiction can be used to show that rx cannot be rational if x is irrational, by dividing both sides by r.
Now that we have that out of the way, the fortunate thing for Algebra students is knowing when the product or sum is rational or irrational. Don't be quick to assume that a radical sign indicates irrational. They love using the square root of 64 or the cube root of 27, both of which are perfectly rational. However, it helps if you can explain why it is so other than to say, "Well, you know, it's obvious." (And, hopefully, it is.)
There's a lot to consider under this standard, so I'll continue with rational exponents, i.e., fractions. What if we wanted to evaluate an expression like this?
We need to recognize that the radical 5 is the same as 51/2, so
The rules for exponents say to multiply the 1/2 and the 4, giving us 52 or 25.
We can take this further. Suppose we had
The cube root is the same as 1/3 power. So
We can evaluate 63 as 6 * 6 * 6 = 216.
One more example: How would we handle
The fourth root becomes the 1/4 power.
Now we can get a little fancy with and deal with the multiplication of two fractions:
One final note: The answer doesn't always have to be a rational number. You may exchange one rational power for another, one root for a different one. Consider:
Problems could contain any combination of roots and improper fractions, which may or may not have a simple rational answer. But keep the calculator handy just in case you need to know the sixth root of 117,649. Showing your work, of course.
The entire standard reads as follows: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
It then has two components, one for Algebra 1, the other for Algebra 2. The Algebra 1 component is Evaluate integers raised to rational exponents (Algebra 1 - V.9). Don't ask me about the "V.9", I've done enough searching for this column.
I used to start teaching each year with Order of Operations, something that all the students should have seen before, and yet seemed to forget about. They know an acronym, such as PEMDAS, but don't know what it means. Oh, they know what the six letters mean, but they don't get the concept. And even when they can explain the concept, when push comes to shove and the pressures on (and they're taking a quiz), you find them calculating from left to right as if they hadn't learned anything. I had to change that when we started welcoming calculators into the lesson (even before we started requiring them). The calculators were down the work for them, so they didn't have to learn it, right? Wrong. I just adapted the problems. I started added more operations within fractions and adding exponents, forcing them to pay attention to what they put into their calculator. For instance, you need explicit parentheses to group things in a calculator because the numerator and denominator are implicitly grouped.
But Common Core changes that. They have to learn it much earlier, so they're ready for Algebra by the time they get to high school. Yeah, right. I'm still doing it. But wait, there's more.
After parentheses, come the exponents. Some students know the concept of exponents, and some just press buttons on the calculator. They know (or they'll learn!) that they are multiplying the factor some number of times. What they haven't seen before is a fraction as an exponent. What do you do when you see a fraction? Hide under the desk, usually. Wait for it to be over. I'm not exaggerating much.
First, I have to teach the concept that exponent 1/2 means take the square root, as stated in the gem. And then see how well they know they're square roots, either with or without a calculator. (I try to get them to learn up to 256, with some success.)
Second, I have to teach them that exponent 1/3 means take the cube root. This usually entails explaining what a cube root is, and where to find it on the calculator. Maybe reviewing what "cubing" means, and possibly through in Volume = length X width X height somewhere, just so I can spiral back to it when that comes around again. Depending on the results, I could try to conquer the concepts of fourth and fifth roots. Seriously, a practice webpage asked me (729)(1/6).
At this point, I'm asking: are they getting it, or are they pushing buttons? I don't mind the button pushing if they understand the concept, because then they'll start recognizing patterns and will be smarter about their button pushing.
Okay, so the next step is the real doozy: explaining exponents of 2/3 or 3/5. The students have to deconstruct the fraction. That is, they need to know that 2/3 = (2)X(1/3), so they need both to square it and then take the cube root. And then I'll suggest that you take the cube root first, so that they're dealing with smaller numbers. Sometimes this makes sense to them.
Finally, there's the kicker: improper fractions. If fractions are Dr. Frankenstein, improper fractions are his monster. They're a whole new level of scary, and the first thing they want to do is turn them into mixed numbers. Or decimals. No! Wait! Stop!
Taking the (5/2) power of 16 is as simple as taking the fifth power of the square root of 16. Okay, read that again like you're a ninth grader.
No, it's not that difficult to do once the concept is learned, but it's something they haven't seen before, and they're learning it earlier. I can't remember exactly when the first time I was required to find a third or fourth root. I was probably using logs to do it. But then, I didn't have the calculators that they have today on their phones. They'll have the answers at their fingertips, but I'll make them show the work so I know they get the idea behind it.
