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Post by uscgvet on Aug 4, 2017 11:22:37 GMT -6
0 = -0
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Post by Deleted on Aug 4, 2017 11:43:42 GMT -6
i = -i i=(-1)*i i2=((-1)*i)2i 2=(-1) 2*i 2i 2=1*i 2i 2=i 2-1=-1 1=1 There is something odd about your method, but it looks like you got the right answer. If you work backwards from what you have (in this case, I believe that would be a better way of developing your proof - see below), I believe you have a procedural error in the bolded part. Order of operations requires that you do what is in the parenthesis first. So, the way it is now, you would have to convert the ((-1)*i) 2 to (-i) 2. That takes you to the third step in the arrows I typed below. You can follow what happens after that. 1=1 -1=-1 i 2=i 2i 2=1*i 2i 2=(-1) 2*i 2i2=((-1)*i)2 ----->> i 2 = (-1*i)(-1*i) ---->> i 2 = (-i)(-i) ------>> i2 = (-i)2 ------>> sqrt(i 2) = sqrt((-i) 2) ------>> i = -ii=(-1)*i i = -i So does anyone know of any real numbers where the reciprocal (negative) value is equal to the original value? There is only one scenario where this occurs. I see what you did. Sorry, I worked out my proof by starting from i=-i and going to 1=1, and forgot to reverse it. Sorry, didn't get a ton of sleep last night  . |
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Post by Deleted on Aug 4, 2017 11:50:32 GMT -6
Here's what I should have put (again, sorry!):
1=1
-1=-1
i2=i2
i2=1*i2
i2=(-1)2*i2
i2=((-1)*i)2
√(i2)=√(((-1)*i)2)
i=(-1)*i
i=-i
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Post by yardstick on Aug 4, 2017 12:20:26 GMT -6
Okay, two scenarios; although I believe mathematically, it is impossible for 0 to be negative.
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Post by uscgvet on Aug 4, 2017 12:47:20 GMT -6
Okay, two scenarios; although I believe mathematically, it is impossible for 0 to be negative. Well... You can chalk board your theory on "impossible" all you want... As a computer scientist, I can actually store your zero as a negatively signed value.  "People Who Say It Cannot Be Done Should Not Interrupt Those Who Are Doing It" --(quote most likely from) George Bernard Shaw |
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Post by yardstick on Aug 4, 2017 13:08:59 GMT -6
Okay, two scenarios; although I believe mathematically, it is impossible for 0 to be negative. Well... You can chalk board your theory on "impossible" all you want... As a computer scientist, I can actually store your zero as a negatively signed value. "People Who Say It Cannot Be Done Should Not Interrupt Those Who Are Doing It" --(quote most likely from) George Bernard Shaw I believe I can grasp why it is necessary for computer programming to have an ability to create a -0: The sign is programmatically independent of the value. However, I think someone would be hard-pressed to find a -0 on a number line... |
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Post by uscgvet on Aug 4, 2017 13:28:43 GMT -6
Well... You can chalk board your theory on "impossible" all you want... As a computer scientist, I can actually store your zero as a negatively signed value. "People Who Say It Cannot Be Done Should Not Interrupt Those Who Are Doing It" --(quote most likely from) George Bernard Shaw I believe I can grasp why it is necessary for computer programming to have an ability to create a -0: The sign is programmatically independent of the value. However, I think someone would be hard-pressed to find a -0 on a number line... Well, I was just being playful...  Please continue this interesting thread. |
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Post by whatif on Aug 4, 2017 15:24:41 GMT -6
Just have to say you all are amazing and wonderful, and I'm loving this thread even while I struggle with simple division and multiplication!  Keep up the great work together! |
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Post by yardstick on Aug 4, 2017 15:28:15 GMT -6
I believe I can grasp why it is necessary for computer programming to have an ability to create a -0: The sign is programmatically independent of the value. However, I think someone would be hard-pressed to find a -0 on a number line... Well, I was just being playful... Please continue this interesting thread. |
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Post by yardstick on Aug 5, 2017 21:02:27 GMT -6
Here's what I should have put (again, sorry!): 1=1 -1=-1 i 2=i 2i 2=1*i 2i 2=(-1) 2*i 2i 2=((-1)*i) 2 √(i 2)=√(((-1)*i) 2) i=(-1)*i i=-i Okay, so, two bonus questions: 1. Does anyone want to take a guess as to the significance of a negative valued number equaling its reciprocal not just in size, but in location also? Because that is what I believe is occurring when we show i = -i. 2. Does anyone have a guess as to when the only time is (and its limitation) when a negative value can equal its reciprocal? And for a little context, typically we only worry about magnitude and direction (size and location) when we are talking about vectors... I specify size and location because the point in space we are identifying with a discrete value is exactly that: a point. A vector is a line (by way of comparison). |
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Post by yardstick on Aug 5, 2017 21:44:34 GMT -6
(continued) Okay, that took a lot less time than I expected, so lets for the moment lay out the rules of exponents, and then do a brief discussion of them next time. Here they are: a m x a n = a m+n (Product Rule - multiplying exponents which have the same 'base' in this case 'a', but different exponents together) (a m) n = a m*n (1st Power Rule) (ab) m = a m x b m (2nd Power Rule) a m
(a/b) m = --- (3rd Power Rule, but b cannot equal 0) b m 1 a -n = --- (negative exponent, also known as the inverse) a ncomplicated example of the negative exponent: a m --- = a m-n (Quotient Rule - dividing exponents with the same base, but different exponents) a nI would like you to focus on the patterns you see. Where do the exponents move to? What happens to the bases? What happens to the exponents? (continued...) And continuing on: so her are some examples of the above rules, that will probably make more sense: product rule: 3 2 x 3 3 = 3 5and if you math this out understanding that exponents are a shortcut for multiplication, you get: (3 x 3) x (3 x 3 x 3) = 3 x 3 x 3 x 3 x 3 right? and 3 x 3 x 3 x 3 x 3 = 3 5 right? 1st power rule: (4 2) 5 = 4 2 x 4 2 x 4 2 x 4 2 x 4 2 right? and that is 4 10 right? for this one, lets do a little substitution to simplify what is going on. let us let the letter b represent 4 2 --> therefore 4 2 = b and flipping it around: b= 4 2 so then when we substitute b for 4 2 we get b 5 right? So b 5 = b x b x b x b x b right? and if we back-subtstitute the 4 2 for the b now, we get ---> 4 2 x 4 2 x 4 2 x 4 2 x 4 2 which is what we showed you above, and is using the product rule to sum the exponent right? 2nd power rule: (4 x 3) 2 Lets do a little review to explain this one. we know that 5 2 = 5 x 5, right? How is 5 by itself represented in exponents? how many 5s are getting multiplied together? If you answered one, you are correct, a single constant (in this case 5) is represented by 5 1 when using exponents. With that knowledge, we can now alter the problem: (4 1 x 3 1) 2 correct? which can also be written like this: ((4 1)(3 1)) 2 if we let the letter m represent (4 1)(3 1), we can do what we just did above and say that ((4 1)(3 1)) 2 = m 2 right? and then we can back-substitute like before: m x m = (4 1)(3 1) x (4 1)(3 1) which is (4 1)(3 1)(4 1)(3 1), and rearranging: (4 1)(4 1)(3 1)(3 1) regrouping (4)(4)(3)(3) = 4 2 x 3 2 which is the solution to the 2nd power rule right? okay. Third power rule: (2/7) 3 lets remember something about fractions: 2/7 = 2 x 1/7 right? its the same thing as working backwards in the multiplication of fractions 2/7 = 2/1 x 1/7, right?  all I did was flip the equation around, like I did before. so since this is true, we can apply the 2nd power rule to this to get  now lets use the quotient rule to explain negative exponents. This will work out nicely as we can use the quotient rule and the negative exponent rule to prove each other:  (continued...) |
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Post by Deleted on Aug 5, 2017 21:58:09 GMT -6
Yardstick, your math comprehension is light years beyond me......even as an engineer once upon a time....
I looked up tesseract, and it still seems in a 3-D XYZ coordinate system that a vector could be drawn in space to each point of each cube or plane, so I still cannot comprehend 4-D! Can you help me out? Why is it not just a fancy 3-D arrangement?
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Post by yardstick on Aug 5, 2017 22:10:48 GMT -6
(Continued) Okay, lets talk briefly about roots. Roots undo exponents, as long as the exponent is not a variable. A variable is a letter we use to represent an unknown, or changeable value (number). the radical sign is used to represent a root. Sometimes there is a small number in the notch of the radical, and that is telling us what degree the root is. if there is no number, that is the default for square root. Here are a couple of examples:  You can see that there is no number in the notch of the radical over the 4. A two is implied.
the radical over the four is read as the square root of four. The radical over the 27 is read as the cube root of 27. Remember how we got an inverse when there was a negative exponent? 1 9 -2 = --- 9 2 well, we represent a root by using a fraction in the exponent:  when using radicals, we are fundamentally doing this:  (it looks a little small, so you may need to open the photo) This is the part where I point out what happens when you get a negative value under the radical: You get an imaginary number. i is the symbol we use to represent an imaginary number. At the most basic level, the square root of -1: (-1) 1/2 = i. another way to express this is i2 = -1. In higher math and engineering, what we are seeing in this situation is that there is a real component and also an imaginary component that are roots of the value under the radical. I'll go into this in a bit. (continued...) |
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Post by yardstick on Aug 5, 2017 22:31:44 GMT -6
(Continued) Ok! next topic! Algebra with polynomials. Believe it or not, once you start learning math, youre doing algebra, just not with the funky letters everywhere: 2 + 2 = ___ 2 + ____ = 5 All algebra is doing is substituting a letter for the ______ The most common variable in algebra is x which is why the times symbol was replaced with a dot, or parenthesized values. 2 + 2 = x 2 + x = 5 Would anyone care to guess what the exponent for the x in these equations is? If you answered 1, you are correct. equations like the ones above allow us to plot points on a one-dimensional line. to plot points in a two dimensional system, we need two dimensions: x + y = 7 we need a value for the x-axis, and a value for the y-axis to solve this problem. what could these values be? Well, first we have to figure out which of these variables is dependent on the other. Dependent means that you have to have a value for the Independent variable before you can figure out the value of the dependent variable. Most of the time, the independent variable is x, and the dependent variable is y. There is also a special way that we represent which variable is dependent, and which variable is independent. We put the dependent variable on one side of the equals sign; and everything else, including the independent variable on the other side of the equals sign. Like this: y = x - 3 Three dimensions require x + y + z = 21, et c. All of the above examples are what is called linear systems because the exponent power of any of the variables is only 1. Linear means 'line' - the two dimensional object we previously discussed. A common template for a linear system in two dimensions is what we call the equation of a line: y = mx + b Where the value of b denotes where the line crosses the y axis if we plot the line on a graph, the value of m denotes the slope of the line. Here is an example: y = x + 9 what is the value of b? 9 what is the value of m? 1 Where do I get the 1 from? well:  right? So we can re-write the equation as y = 1x + 9, showing the slope is 1 and the y-intercept (where the line crosses the y-axis if we plot it) is 9. This tells us that the value of y is going to change at the exact same rate that the value of x changes. (continued...) |
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Post by watchmanjim on Aug 5, 2017 22:51:20 GMT -6
X-x
~
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Please continue this interesting thread.
Keep up the great work together!
