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Post by sawdy on Aug 3, 2017 8:23:29 GMT -6
Even though there is only one cube god, they come to the conclusion that there are three gods, because the god each saw appeared differently to each person in their two-dimensional world. Therefore a god that exhibits extra dimensions, when described in a lesser dimensional universe would seem to represent more than one entity. Likewise, our God, who must exist and operate in dimensions beyond our understanding, exists as a Trinity ] Thanks for this, I am going to use this to help explain the concept of the Trinity to my children. |
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Post by uscgvet on Aug 3, 2017 8:41:11 GMT -6
Would anyone like to take a guess at what a 0-dimensional object is called? I would say this is a location or point in any and/or all of the dimensions. Including the time dimension. I wouldn't call it an object though as it lacks dimension in itself. |
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Post by Deleted on Aug 3, 2017 9:20:50 GMT -6
Hey, yardstick, have you ever done any research into Hilbert spaces? I know it's a vector space where thr vectors are square integrable functions. CAN Can you explain that intuitively?
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Post by Deleted on Aug 3, 2017 11:18:05 GMT -6
Oh yes, one more question yardstick. Can you PLEASE teach me how to understand and use tensors? They are ridiculous!
Actually, if you have any good resources on diffrential geometry, preferably riemannian geometry, that would be nice.
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Post by Deleted on Aug 3, 2017 11:19:03 GMT -6
Well, correction. I can use tensors to calculate the Christoffel symbols from a given metric tensor, but I'd like to know how to understand them and how to use them in other contexts.
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Post by delo63 on Aug 3, 2017 11:33:05 GMT -6
I hope im not the only one who's lost lol
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Post by Deleted on Aug 3, 2017 13:00:49 GMT -6
My questions are out there, I know LOL! Yardstick seems very capable and I've never had anyone to ask these type of questions to, so I'm asking him.
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Post by whatif on Aug 3, 2017 14:27:09 GMT -6
Isn't it awesome how God provides?
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Post by yardstick on Aug 3, 2017 18:38:02 GMT -6
(continued)
Okay, I see a lot of questions, some of which I can answer, and some of which I would have to do some research on before I could answer whether I could answer; and at least one where your knowledge level is greater than mine: Tensors- I do not recall doing coursework in those, sorry. the Calculus related to Reimann stuff I did a little of, and can try to dredge it up, and i can definitely show a little of differential geometry (i think).
1. a 0-dimensional object is a point, which does not have size, but does have locality. Kudos to Beloved who got that right, and to others who rightly commented about details of 0-dimensional objects.
2. For everyone who feels a little overwhelmed or behind, please be patient, because I am going to give you some applied math (real world use) and I think as soon as you see it, and the real world application, you'll grasp the concept, even if the formulas make NO sense at all!
3. Socalexile, your insertion of the explanation of multidimensional objects was very cool, but actually more sophisticated than I was going to use. I really liked the 'cube-god' analogy. That was amazing. So was the Mrs. Flat hiding in the box example. Made me reconsider my hypothesis about the Mark giving making people give off pheromones. That becomes unnecessary if you have a 10-dimensional fallen angel looking for your 3-dimensional person. The angel can just see you anywhere.
4. I was thinking about this thread a lot earlier today and came to the reluctant conclusion that I am going to have to cover a little algebra, trigonometry and calculus in order to explain 5th-7th dimensional stuff, and going to have to do a proof to demonstrate 8, 9, and 10 dimensional stuff too. And quite honestly, hope that I am right in my understanding, as my hypothesis is that 5th-10th dimensional math requires the use of imaginary numbers. I may also have to dabble in the notion of 'discrete' values in the sense of 'analog' versus 'digital'.
(continued...)
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Post by yardstick on Aug 3, 2017 19:15:11 GMT -6
(continued)
This is where we start wandering away from the initial concept where the thread idea came from, because it is necessary to provide some underlying but essential mathematics in algebra, trig and calculus, followed by some practical application of each. Also necessary is a brief explanation of imaginary numbers, so lets start with that.
We first learn to count numbers, which as we count up or down, infers the relationships between numbers. All numbers are related to each other. Example by way of a question:
What is the relationship between 3 and 4? The answer is 1. Whether you are counting from 3 to 4 or going backwards from 4 to 3, you are moving 1 unit on a one-dimensional number line.
As we move from number to number, we identify those as discrete values. Remember the term discrete. I'll be referencing it again later. 27 is not the same as 27.00123545 nor is it the same as 27.1. It is in fact 27.00000~ where the zeroes are infinite.
When someone asks us the time, and we say, "Oh, its 3:30." when the digital clock reads 3:27, or the analog clock (the one with hands) has the hour hand pointing somewhere in between the 3 and the 4, the minute hand pointing somewhere in between the 5 and the 6, and the second hand either clicking its way around discretely (no pun intended) or constantly revolving around the centerpoint. When we made the statement, did we lie? Or did we round the time to the nearest half-hour? The constantly moving hand on a handed clock is what makes the clock analog. a digital clock has discrete values for the time. An analog clock does not - all of the hands are usually in between numbers, right? How do you measure that? Because however you measure it, you cannot get a precise value since at any given point in time, the hands (particularly the constantly moving second hand) are in between numbers. You have to round - You have to discretize the value that the clock is giving you for a time! And when you do this, you introduce error into the value.
Remember when you were taught rounding of decimals in school? How about 'significant figures'? "Sig-figs" is the rule for how to round to minimize discretizing error!
Okay, sorry bout that, went off on a 'tangent'. Lets get back to numbers...
So after we learn the numbers, and learn to count, we begin to learn to add numbers. Remember when I explained the relationship between any two adjacent numbers is 1? What if they are not adjacent? What if you are trying to find the relationship between 2 and 7? How do we do that? We accumulate the relationships of each of the adjacent numbers from 2 to 7. The shortcut terms we use for that is addition and subtraction.
2+___ = 7 right? where the ____ is the accumulation of the the relationships between both numbers. We define that accumulation as either addition, or subtraction, depending on whether we are moving along the number line from left to right (addition) or right to left (subtraction).
So so we learn the shortcut for counting long distances of numbers is addition and subtraction.
There is a shortcut for addition also. Yes, multiplication. But before we go on, I need to make a simple point that is a bit difficult to comprehend. There is no such thing as subtraction. Wait, what? Did you notice where I bolded left to right and right to left up there? When we 'add' numbers, we are moving along the number line from left to right. When we 'subtract' numbers we are moving along the number line from right to left. The terms addition and subtraction simply tell us which direction to move!
Subtraction is a construct, a paradigm, a way of thinking, maybe even a method. By way of example, lets take a problem you have probably seen before:
-3 - 3 = -6
How did we know that the answer is -6? Map it on the number line! Start at the first -3 and move +3 units to the left! You arrive at -6! Re-writing the equation we can visualize it like this:
-3 + (-3) = -6
Right? Did we do any 'subtraction' in that last equation, or did we do addition? If we didn't need subtraction to get the solution, we find that the idea of subtraction is redundant and unnecessary. Looking at it another way:
-3 - (+3) = -6
The second negative sign tells us to move left on the number line! Did we really 'take away' anything?
And so now I am going to blow your mind again by telling you that there is no such thing as division, for the same reason! Division is the inverse of multiplication. I will demonstrate this when we get to fractions.
OK, lets move on.
Addition is the shortcut for counting, and Multiplication is the shortcut for addition. How tedious would it be to add this together:
2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2+2
That's a lot of twos!
Wouldn't it save ink, paper, time and our carpal tunnel to simply write:
2 x 24 ?
Similarly exponents are shortcuts for multiplication by the same method:
2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 is the same as 218
(continued...)
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Post by yardstick on Aug 3, 2017 20:00:34 GMT -6
Okay, so we talked about how numbers, and counting are referring to discrete values on a number line, how addition tells us to move from left to right on the number line, and if we must use the term subtraction, we really only need it to tell us that we are moving from right to left on the number line. We also discussed that adding is a shortcut for counting, multiplication is a shortcut for adding, and exponents are a shortcut for multiplication. This is the part where I demonstrate that there is no such thing as division. But before I do that we have to re-visit fractions. There are three terms I want you to remember about fractions, when we are describing what type of fraction we are working with: Proper fractions, improper fractions and mixed fractions. - Proper fractions have the smaller number on the top - 2/3
- Improper fractions have the smaller number on the bottom - 3/2
- Mixed fractions have a whole number (discretized) added to a proper fraction - 1 2/3 (formatting appears to be a challenge in this environment, so I am going to denote mixed fractions like they do in construction: with a dash between the whole number and the fraction and no spaces between it and the numbers... 1-2/3)
Also, the top number is called the numerator. The bottom number is called the denominator.
We can convert from improper fractions to mixed fractions and back again. To convert from mixed to improper, multiply the whole number by the number in the denominator, and then add the numerator. place the final value over the original denominator. Example:
12-7/8 = 103/8 -------->> 12 x 8 = 96 +7 = 103 ---------->> 103/8
To go the other way, from improper to mixed, determine how many multiples of the denominator will fit into the numerator without going over. This becomes the whole number. The remainder, which will be a value less than the denominator, becomes the numerator of the mixed fraction. Example:
173/13 ---------->> 13 x 13 = 169, remainder is 173 - 169 = 4, therefore, mixed number is 13-4/13.
Aside: I found four things as a tutor, that people who were going to college and taking college level math (including algebra) were weak in:
Multiplication Tables
Factoring
Rules of Fractions
Rules of Exponents
To do the first level of calculus, you have to know these four things, plus logarithms. We will not need logarithms for the 'math class' here, so I will skip them, but rules of exponents are huge. Especially in the higher levels of calculus. Anyhow, lets cover Rules of Fractions first. You need to know two basic rules. How to add/subtract and how to divide/multiply. That's it. everything you do with fractions in it will use those two rules. We are going to delve a little into abstractions of algebra to demonstrate the model, but will follow with a practical application.
Rules of Fractions: adding and subtracting
When multiplying fractions you need to look for this pattern:
 here is a 'numeric' example of the pattern:  The 'dot' in the first photo is just a shortcut for the usual X we use in grade school. They mean the same thing. So we solve this simply by multiplying the numerators together, and multiplying the denominators together. The result (product), is 6/28. Now, we didn't discuss reducing fractions, which is a form of simplifying them, but you would do that by finding a number in your multiplication tables that both 6 and 28 have in common (its a factor of each called the greatest common factor). This is where the factoring comes in: 2 x 3 = 6 and 1 x 6 = 6, so the factors of 6 are: 1, 2, 3, 6. Similarly 1 x 28 2 x 14 4 x 7 all equal 28, so the factors of 28 are 1, 2, 4, 7, 14, 28. If you compare both lists of numbers, which number is the one that is both common to both lists, and is the largest common number? If you answered 2, you are correct. Both lists have a 2, and 2 is the largest number that both lists have in common. You could also use the smallest number that both lists have in common to reduce the 6/28 fraction, if there was one, but you would have to do this exercise more than one time. so we take the 2 and divide both the numerator and the denominator by them: 6/2 3 --- = --- 28/2 14 Neither 3 nor 14 have a common factor, so the fraction is as reduced as it is going to get. Division of fractions. Remember I said there was no division, and that division was just an inverse of multiplication? This is where that statement comes from. It turns out that division of fractions is exactly the same as multiplication, but with one additional step. The inverse step. The pattern for division is almost identical to multiplication, except there is a division sign there.  Like the multiplication example, we are going to use the same numbers, but divide the first number by the second one. And like the multiplication example, we are going to multiply the denominators together, and multiply the numerators togeher, then ultimately reduce the fraction if necessary. But there is one twist. Before we can multiply the fractions together we must first invert the second fraction! Flip it upside down. (continued...) |
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Post by yardstick on Aug 3, 2017 21:22:03 GMT -6
So lets set up the example, then invert the second fraction:   As you can see, when we invert the second fraction, we also have to change the mathematical operation from division to multiplication. This is why i said there was no division. All I have to do is invert and multiply! and when we do that we get: 8/9, which is not reducible because 8 and 9 do not share a common factor. Okay, I will pick up with adding and subtracting next time, and try to get through a bit more also. by demand I need to see if I can get through the last of the fractions so I can cover exponents tomorrow, linear algebra and non-linear algebra Saturday and Trig Sunday. I am dwelling on these topics because mathematics is structured like a stairway. If you do not have the lower stairs firmly in place (or they are missing) you cannot get to the higher levels. So Adding and Subtracting fractions has a pattern also:  You can substitute a - sign for the + sign and it works the same. The key take away on adding and subtracting fractions, is that, unlike multiplying and dividing, you must have a denominator that is the same for one fraction as it is for the other. (continued...) |
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Post by Deleted on Aug 3, 2017 21:29:33 GMT -6
Dang your starting from scratch!
I know this probably ain't helpful now, but to all y'all who know the beginning stuff, an imaginary number is defined by a multiple of the square root of negative 1.
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Post by yardstick on Aug 3, 2017 21:36:28 GMT -6
Dang your starting from scratch! I know this probably ain't helpful now, but to all y'all who know the beginning stuff, an imaginary number is defined by a multiple of the square root of negative 1. I am, and I apologize, but I did notice that there are a few people who I lost early on. I also know that mathematics is one of those things: you lose it, if you dont use it. Thing about math is, as you go higher, it gets more and more abstract. To get to square roots, One needs to recall natural squares of their multiplication tables. To do natural squares, one must know exponents. Exponents is next after fractions. Notice I basically skipped factoring, except a brief description. I am also only covering the two primary rules of fractions related to addition and subtraction, and multiplication and division. Even though I wanted to skip factoring and mixed fractions, I had to cover them briefly when they interacted with the addition and multiplication parts. You have to convert mixed fractions to improper in order to have them in the correct form for multiplying them. Also, what happens when we swap out the real numbers here for abstract algebraic symbols? I can promise you that as soon as I get through rules of exponents, I am going to be covering Linear equations, polynomial equations and Trigonometry. I expect to start hitting the 'meat' of this topic this weekend. |
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Post by yardstick on Aug 3, 2017 22:10:05 GMT -6
(continued) Okay, so an example of adding or subtracting fractions, which requires them to have the same value in their denominators would be like this:  The first thing we have to do is get the denominators to be the same value. To do that, we have to get what they call the least common multiple from the multiplication tables: mutiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, et c. multiples of 7 are: 7, 14, 21, 28, 35, et c. As you can see, the multiple that each has in common that is the lowest one is 28. So that is what we need to have in the denominator. But we cant just change the denominator to 28! We need to change the appearance of the fraction without changing its value.What I mean is that 5/28 is not the same value as 5/7, right? So to get the same value with a different appearance, we need to perform a mathematical trick. We need to manipulate the fraction without changing its value: we do it like this: 4/4 = 1 -------> 5/7 x 1 = 5/7, --------> 5/7 x 4/4 = 20/28 similarly, 7/7 = 1 ---------> 3/4 x 1 = 3/4 --------> 3/4 x 7/7 = 21/28. Now both fractions have the same denominator and we can add them: 20/28 + 21/28 = 41/28. Did you notice that I didn't add the denominators? For adding and subtracting we convert the denominators to the same value and then retain that value as the denominator for the sum of the numerators. Also, I could convert the improper fraction to a mixed fraction; however mixed fractions are more difficult to work with in the upper math levels, so generally we just leave improper fractions improper. (continued...) |
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