### Rationalization

By now I have taught easily a dozen freshman-level college math courses. In every single one that I have taught I have faced the brainwashing that students have received to rationalize the denominator of a fraction. (I use the word "brainwashing" here loosely. Clearly I do not accuse any high school math instructor of actually using brainwashing techniques.) For those that aren't aware or don't remember, a denominator is irrational when it has a square root (or any other root in it). For example, 1/√2 has and irrational denominator. The denominator can be rationalized by multiplying this fraction by 1 in the form of √2/√2, thus making it √2/2.

The first thing that I find extremely ironic about this is that there are some fractions that simply cannot be rationalized. This occurs when a transcendental number is in the denominator, such as π. But, that aside, there is no mathematical reason for rationalizing the denominator. There is no real benefit in doing so. Surely, there is no harm in it. It does not damage the number in question, since it is simply another way of writing the same number. But it is completely useless. It serves no actual purpose. The fraction √2/2 is in no way superior to or more useful than 1/√2.

In classes where I teach trigonometry (usually pre-calculus), I make a point to tell the students that this is the case. I instruct them that there is nothing wrong with having an irrational denominator. I use fractions that have irrational denominators to get them used to the idea. And all throughout the semester, my students have problem with this. The cognitive dissonance is obvious. They believe that the instruction from their high school teacher(s) to rationalize the denominator is just as important as any other instruction they received in math class. In the hopes of minimizing this cognitive dissonance, I do tell them that if they prefer, they are welcome to rationalize denominators since it poses no threat to mathematical rigor.

The only reason I can think that this concept has permeated high school mathematics education is that there are methods in calculus for finding particular limits where either the numerator or the denominator are rationalized in order to solve the problem. But, whatever the reason, the fact remains that this is regularly taught in high school. In fact, my own algebra and calculus teacher taught me that I should rationalize denominators. For a time, I actually believed that there was a reason for this. Then, I went to college and discovered that it's completely meaningless.

So, if you're reading this post you're probably thinking to yourself "This can't really be about math, so what are you getting at?" You're right. This is an analogy. Just as rationalizing the denominator is harmless and yet completely useless, so too are many religious practices. And just as I, and most college math instructors, work really hard to undo the brainwashing done by high school math teachers, so too do atheists work hard to undo the brainwashing of religious leaders.

Consider, for example, the religious practice of prayer. Unless you mean to discuss a really radical religion or really strange prayer practices, this ritual is completely harmless. Normally it consists of one or more people together with one or all of them speaking or thinking words in their head, expecting that these words will reach a deity who will then perform some action based on the prayer. In all of my years as a believer, in all of the thousands of prayers that I have participated in, I have yet to see a single one which has any effect other than to comfort people. Surely, there is no harmful effect of prayer. However, there isn't any effect at all. It's merely a waste of time. Just like rationalizing a denominator, you're welcome to do it, but it's meaningless.

The same principle goes for any religious ritual (okay, maybe human/animal sacrifice or suicide bombing is a different story). There is no real harm that is done in baptism, communion, or whatever other rituals you engage in. But there's no benefit either. There may well be psychological effects, such as comfort for those who pray. But all of these effects can be granted through other means. If I am ill, I can be comforted much more by someone caring for me (in a real sense--making soup for me, etc) than for someone praying for me.

In all my years as a believer, I have witnessed several priesthood blessings--which is really just another form of prayer--and I have even given several myself. And every single one has again had no real benefit other than possibly to comfort the recipient or others involved. I never saw a person healed. I never saw a person's recovery quickened. I heard plenty of stories of such things happening, but never met anyone to whom it had happened.

So, I say to all of you who are religious the same thing I say to my math students. Go ahead and continue your religious practices. They don't harm anything at all. But, they are completely useless and ineffective. You'd save yourself a lot of time by refraining, and you'd be no worse off, just as you save time by not rationalizing denominators. You can know that I'm just trying to be friendly and help you out because I'm not trying to force you one way or the other, I'm just telling you like it is. I'm just trying to save you time, and to help you see the truth.

Yes, you can get comfort from prayer, faith, and priesthood blessings and things like this. But you can get comfort through other means as well. You can find comfort in truth. When I have a headache, I can find comfort in ibuprofen. When I have a hardship or a trial, I can find comfort in knowing that it will help make me stronger and that other people more experienced and wiser than myself have been through such things and may be able to help me. I believe that truth is always the best way to face life. Be honest with yourself and with others. Learn the truth, accept it, and be freed by it.

The first thing that I find extremely ironic about this is that there are some fractions that simply cannot be rationalized. This occurs when a transcendental number is in the denominator, such as π. But, that aside, there is no mathematical reason for rationalizing the denominator. There is no real benefit in doing so. Surely, there is no harm in it. It does not damage the number in question, since it is simply another way of writing the same number. But it is completely useless. It serves no actual purpose. The fraction √2/2 is in no way superior to or more useful than 1/√2.

In classes where I teach trigonometry (usually pre-calculus), I make a point to tell the students that this is the case. I instruct them that there is nothing wrong with having an irrational denominator. I use fractions that have irrational denominators to get them used to the idea. And all throughout the semester, my students have problem with this. The cognitive dissonance is obvious. They believe that the instruction from their high school teacher(s) to rationalize the denominator is just as important as any other instruction they received in math class. In the hopes of minimizing this cognitive dissonance, I do tell them that if they prefer, they are welcome to rationalize denominators since it poses no threat to mathematical rigor.

The only reason I can think that this concept has permeated high school mathematics education is that there are methods in calculus for finding particular limits where either the numerator or the denominator are rationalized in order to solve the problem. But, whatever the reason, the fact remains that this is regularly taught in high school. In fact, my own algebra and calculus teacher taught me that I should rationalize denominators. For a time, I actually believed that there was a reason for this. Then, I went to college and discovered that it's completely meaningless.

So, if you're reading this post you're probably thinking to yourself "This can't really be about math, so what are you getting at?" You're right. This is an analogy. Just as rationalizing the denominator is harmless and yet completely useless, so too are many religious practices. And just as I, and most college math instructors, work really hard to undo the brainwashing done by high school math teachers, so too do atheists work hard to undo the brainwashing of religious leaders.

Consider, for example, the religious practice of prayer. Unless you mean to discuss a really radical religion or really strange prayer practices, this ritual is completely harmless. Normally it consists of one or more people together with one or all of them speaking or thinking words in their head, expecting that these words will reach a deity who will then perform some action based on the prayer. In all of my years as a believer, in all of the thousands of prayers that I have participated in, I have yet to see a single one which has any effect other than to comfort people. Surely, there is no harmful effect of prayer. However, there isn't any effect at all. It's merely a waste of time. Just like rationalizing a denominator, you're welcome to do it, but it's meaningless.

The same principle goes for any religious ritual (okay, maybe human/animal sacrifice or suicide bombing is a different story). There is no real harm that is done in baptism, communion, or whatever other rituals you engage in. But there's no benefit either. There may well be psychological effects, such as comfort for those who pray. But all of these effects can be granted through other means. If I am ill, I can be comforted much more by someone caring for me (in a real sense--making soup for me, etc) than for someone praying for me.

In all my years as a believer, I have witnessed several priesthood blessings--which is really just another form of prayer--and I have even given several myself. And every single one has again had no real benefit other than possibly to comfort the recipient or others involved. I never saw a person healed. I never saw a person's recovery quickened. I heard plenty of stories of such things happening, but never met anyone to whom it had happened.

So, I say to all of you who are religious the same thing I say to my math students. Go ahead and continue your religious practices. They don't harm anything at all. But, they are completely useless and ineffective. You'd save yourself a lot of time by refraining, and you'd be no worse off, just as you save time by not rationalizing denominators. You can know that I'm just trying to be friendly and help you out because I'm not trying to force you one way or the other, I'm just telling you like it is. I'm just trying to save you time, and to help you see the truth.

Yes, you can get comfort from prayer, faith, and priesthood blessings and things like this. But you can get comfort through other means as well. You can find comfort in truth. When I have a headache, I can find comfort in ibuprofen. When I have a hardship or a trial, I can find comfort in knowing that it will help make me stronger and that other people more experienced and wiser than myself have been through such things and may be able to help me. I believe that truth is always the best way to face life. Be honest with yourself and with others. Learn the truth, accept it, and be freed by it.