- #36
Mark44
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OK, we are able to divide a whole (1) into three equal parts that add to 1. IOW 1 = 1/3 + 1/3 + 1/3. Sounds like you're comfortable with that, even though when you convert the fractions to decimal representations, you get something shy of 1.WW_III_ANGRY said:Mark, thank you for bearing with me :)
Let me continue with my understanding in hope that you can see what I am thinking and where I am wrong:
1/3 is able to be represented precisely in a fraction form. Essentially its a piece divided into three parts. Makes perfect logical sense to me and is comprehensible as well as rational and a rational number. I don't see how the comparison as applicable because my dilemma is with a square root which is reducible to fractions or decimals in rational numbers and is also comprehensible and rational to do so if needed. As I see it everything we know of physically can be divided as fractions or decimal expansions, or what have you. As I conceptualize square roots these too must follow the same path logically or in the case of irrational numbers these aren't applicable.
Let's take it a step further. With [itex]\sqrt{2}[/itex], we are dividing 2 into two equal parts that multiply to make 2. IOW, [itex]\sqrt{2}[/itex] * [itex]\sqrt{2}[/itex] = 2. The main difference here, as I see it, is that the decimal representations are not easily predictable. For example, it's clear that the digit in the 25th decimal place of the decimal representation of 1/3 is 3, but although I can find the digit in the 25th decimal place of [itex]\sqrt{2}[/itex], it will take me a bit of work and a lot of paper to do it.
However precise a decimal representation of [itex]\sqrt{2}[/itex] you need, I can find enought decimal digits of [itex]\sqrt{2}[/itex] to satisfy that precision. If we're talking about light years, as you were in a previous thread, all I need is about 13 or 14 digits of [itex]\sqrt{2}[/itex] to have a precision within a mile. If you need more precision than that, I'll just do some more calculations and get more digits.
What's more, using nothing more than paper and pencil, I can supply you an approximation of [itex]\sqrt{2}[/itex] that is so close to the actual value that if you squared it on your calculator (assuming you have one), the result would be 2. That's just plain 2, with no decimal point and no digits after the 2. I would have "fooled" your calculator by giving it a value that's not quite equal to [itex]\sqrt{2}[/itex]. The calculator would multiply the value I supply by itself and come up with a result so close to 2, that the calculator would be unable to distinguish the result from 2.
1/3 is more desirable computationally as well, since using .333 in place of 1/3 results in errors. If you want smaller errors, use a representation that is more precise. Same with using a decimal representation of [itex]\sqrt{2}[/itex]: the more digits in your representation, the better your calculated result. What's so incomprehensible about that?WW_III_ANGRY said:I understand geometrically that 1/3 is more desirable than .333, however as stated above this is able to be comprehended conceptually and physically. I understand the square root of 2 is able to be utilized to find the hypotenuse of a right triangle, however that is an incomprehensible value to me as well as it is unable to be expressed in the physical, or conceptually through decimal expansion or fraction. Which is why I see it as irrational in the whole sense of the word :)
Even ordinary decimal fractions don't always work the way we think they should when we start doing calculations with calculators and computers. For example, 1/10 = 0.1 exactly, but in some programming languages (such as C and languages based on C), if you add this number to itself 100 times (meaning that you have an addition problem with 100 terms, all of which are .1), you don't get exactly 10. The reason for this is that 1/10 has a very nice, short decimal representation in base 10, but computers usually work in different number systems, base 2 or base 16, and conversion from one base to another can lead to surprising results. So how comprehensible is that?
The upshot of my rambling here is that a significant chunk of mathematics is concerned with approximations, and how close the approximations are to what they are supposed to represent. Like or or not, most numbers (by far) are irrational, so the best we can do when we need to calculate something with them is to use as much precision in our representations as is required to get the results we're after.