Tuesday, August 26, 2014

N-RN.3 (Real Number System) - Irrational Behavior

A short entry for tonight.

Moving on to another Common Core Algebra standard brings me to N-RN.3, which reads

Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

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.)

1 comment:

Bill Ricker said...

perhaps the hardest part at this level is that x*y, both irrational (in R\Q) can sometimes be rational algebraic or even integer.

This is an "obvious" (to whom?) corollary of your warning that there are rational roots. E.g., sqrt(8) and sqrt(2) are irrational but √8√2=√16=4, and √½√4.5=√(9/4)=3/2=1.5

Which means if you know one factor is irrational algebraic but don't know about the other, you have no clue what class the product is.

(And then there's the transcendental numbers, even more irrational and contagious than the algebraic irrationals, but still capable of cancellation. Euler's Identity is beautiful ! But that's probably in a higher tier Regents/CC syllabus than Algebra 1.)