After twenty-two years of mathematics teaching, the urge to explore and explain mathematical concepts has become a force of habit. Recently, I was exploring Mathcad’s equality symbols and I came up with an idea for an algebra lesson on numerical inequality that leverages Mathcad’s powerful computational thinking and communication capabilities to dig deeper into the concepts of equality and rational numbers.
Exploring equality with the Mathcad’s Numeric Evaluation Symbol (=)
Consider the expression:
If we were to estimate its value, we would expect a quantity greater than 5 (25/5), but less than 6.25 (25/4). What results does Mathcad give us?
Using the Mathcad’s numerical evaluation equal sign:
Given our estimate that seems about right. We could satisfy ourselves with that, but let’s check the result. Using Mathcad’s Boolean Equal sign:
Well, this is interesting. The result of 0 that Mathcad gives us let’s us know that this Boolean equation is false. What is going on here?
Perhaps we have a rounding error? Maybe we should extend the evaluation to 6 decimal places?
Well, that did not fix the problem
Are we concerned that our answer is wrong? Probably not. But, what would it mean to be correct?
Using Fractions instead of Decimals
Let’s try something different: Let’s convert the result to a fraction.
Well, this is another surprise! 8,377,962 over 1,571,845!!! Did something go wrong? How would we check:
So, it seems like we are on track. But, let’s use this fraction in a Boolean statement.
It’s still false.
Gee, we are really getting deep into the mud of concept of equality today. Like a field trip, really.
Try a new approach – Use Mathcad’s Symbolic Evaluation Symbol (->)
Let’s use a different equality symbol. Mathcad has a second evaluation operator to supplement the = evaluation symbol. The -> evaluator is the symbolic evaluation symbol. Just to illustrate the difference, here’s an example:
The arrow gives us a symbolic result that is algebraically equivalent to the given expression. The numeric evaluator gives us an error in the first instance because x was not defined. But, after using the assignment operator (:=), both the numeric and the symbolic evaluation symbol return the same result.
Now, what happens if we evaluate our tricky numerical expression with the symbolic evaluation operator.
Wow! The numerator was simplified as 17 + 8 = 25. And, the denominator was rationalized — Mathcad removed the radical number from the denominator by multiplying the top and bottom of the fraction by the square root of 22.
Yet, even this result is not truly equal to the original expression.
So far, in evaluating the original expression we have determined that
- Is between 5 and 6.25
- Is close to 5.33
- But, a more exact estimate is 5.330018
- Two fractional results that estimate the exact value of the expression are
- And, none of these quantities is exactly equal to the original expression in Mathcad.
Are we stuck? Is the exact value unknowable?
Why all the inequality?
By now, we should have realized that the expression we are working with is an irrational number like pi and therefore we will never know the exact answer. But, what does Mathcad think? Is Mathcad teasing us, rather than just telling us that the expression is irrational and all answers are estimates?
No, Mathcad is evaluating the expression as exactly as it can and then assessing our Boolean statements against its best estimate. Let’s see what happens if we maximize the number of decimal places in our result.
Finally, a true statement! But, is this the exact answer? Not entirely, because the quantity is still irrational. But, we are now within one-one-hundred-trillionth of the precise answer. And, that is the default limit on Mathcad’s numeric calculations.
So, even with a powerful computational tool like Mathcad, many of our results are approximations. And this is a grand argument for using tools like Mathcad in secondary mathematics. How else would we dig so deep into the concepts of equality and irrational numbers? How else will we prepare our secondary students for careers in STEM fields where tools like Mathcad are the industry standard for engineering and scientific calculations?
Blogger’s Note: The text and equations were originally completed in Mathcad – as if presenting to a class. In Mathcad, the mathematical statements would be live, but that capability is lost in translation to Word. For those who have Mathcad, if you would like the original Mathcad 15 file email me at MathcadEducation@ptc.com. If you wish download a free trial of Mathcad, you can do so at Mathcad Free Trial Download.