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| William McCormick |
 Posted: Thu Apr 10, 2008 5:20 pm Post subject: Out Of Order! |
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Joined: 03 Apr 2008
Posts: 388
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I think that the reason today that most do not do more with math, is the entry system and order of math. Today the standard is a chaos in my opinion.
For many years great mathematicians worked on ways to enter numbers into a computer so that there would be no out of order thinking, for example adding parenthesis around numbers first before you type the number.
I don't know about you but, I don't think about math like , parenthesis 2+2 times four. Or four times parenthesis 2+2.
I learned math in a place that at the time was considered the math capital in the World. I worked on a Main Frame computer they bought in 1971 for one million dollars. I was a mathematician and honor student in the fifth grade.
Not very good but I had a stubborn attitude. My point is that we learned an in line order entry system that was well ahead of its time and allowed anyone that knew the system to pretty much understand the equation.
It allowed easy entry and no parenthesis, in almost all cases of entering just about any equation you could come up with.
The order of the computer parsing the equation went, addition, subtraction, multiplication and division. There were a couple exceptions to this.
The small case "x" was to act much like a fraction symbol at the time acted. In other words when you see a fraction with the "/" symbol separating two numbers you take it as a fraction.
In this entry system the lower case "x" did the same thing it isolated the numbers just to the left and right of the lower case "x". The Upper case "X" had the same meaning as the division symbol at the time. The division symbol, "÷" meant to take everything on the left and divide it by everything on the right. The Upper case "X" meant the same thing. To take everything on the left and multiply it by everything on the right.
In this way you could perform almost all calculations with no parenthesis. Most common input equations used in real life, work fastest and best like this.
Today we put in a lot of x,y coordinates. The "x" of computer designed input would probably conflict with many formulas today.
You could enter fractions and not enclose them in parenthesis. Fractions are highly overlooked for their accuracy and ease of manipulation.
An example, we have three separate parts to whole, to be added up, twenty three, plus fifteen, plus fourteen and a half multiplied by five and divided by four.
Written in computer optimized format. This is a very real common input or would be. 23+15+14 1/2X5÷4 The answer would be 65.
Normally I have to turn that into a super complex formula like this. ((23+15+14) +(1/2))x5÷4
Can you see how I don't think like this or talk like this? I don't say parentheses parentheses 23 plus 15 plus....................
Here I have a space that I want to divide into four separate compartments. I need to know the compartment size so I can place the dividers.
The dividers are 3/4 inches wide. The total space is 45 inches.
The formula would be 45-3/4x3÷4 So the answer would be 10 11/16"
This is a very common formula and to write it out in what is now the standard method is a little bizarre. 45-((3/4)x3))/4, it is things like this that we the working individuals do to make anything possible on earth. To have to express it the way we do now, is just plain non-mathematical.
IBM could not do it at the time. So they opted to leave the in line division key off the keyboard.
Here is a good one. The front side of a building is made up of 10 identical segments of 14 feet each plus two 13 foot segments on each end and one 12 foot center section that divides the front of the building. The side of the building is made up of three identical segments 10 feet long that over hang by two feet on each end. I want to know the area of the building, it is a square building. Here is the formula that was designed for computer input. 10x14+2x13+12X3x10-2-2=
Here is that formula to enter into a calculator just to get the area. (10x14+2x13+12)x(3x10-2-2), wow, do you see why I get a little depressed by current math?
I was supposed to be able to talk and enter the formula into the computer. Now to me we have chaos.
Try my system out honestly even if it is not easy at first give it a real shot.
Sincerely,
William McCormick
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| serpicojr |
| Posted: Thu Apr 10, 2008 6:30 pm Post subject: Re: Out Of Order! |
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| William McCormick wrote: |
| I learned math in a place that at the time was considered the math capital in the World. |
William, really, Freeport, NY, was never the math capital of the world.
And, William, you've got to get off of this idea that math is somehow hampered by parentheses. It's not. |
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| Harold14370 |
| Posted: Thu Apr 10, 2008 6:46 pm Post subject: Re: Out Of Order! |
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| William McCormick wrote: |
Here is the formula that was designed for computer input. 10x14+2x13+12X3x10-2-2=
Here is that formula to enter into a calculator just to get the area. (10x14+2x13+12)x(3x10-2-2), wow, do you see why I get a little depressed by current math?
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No, I don't. |
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| William McCormick |
| Posted: Thu Apr 10, 2008 7:33 pm Post subject: Re: Out Of Order! |
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Joined: 03 Apr 2008
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| Harold14370 wrote: |
| William McCormick wrote: |
Here is the formula that was designed for computer input. 10x14+2x13+12X3x10-2-2=
Here is that formula to enter into a calculator just to get the area. (10x14+2x13+12)x(3x10-2-2), wow, do you see why I get a little depressed by current math?
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No, I don't. |
You are used to your way of doing it.
Imagine though if you actually created accurate mechanical shop drawings all the time. All you do is take the actuality and turn it into cyber reality. For a computer generated reality that you build in. You do not want to use parenthesis trust me.
The drawings have to be extremely accurate, or you get some wild errors. And a bunch of garbage.
I am saying that I have to think and enter in parenthesis all day sometimes. It is like having a gold cup but only being allowed to drink from a wet brown cardboard cup.
But I can think and enter things exactly as they are into the computer for comparison, alteration and an exact view of what alterations or creations will look like. All to scale.
I work with a pretty good program now that does allow for some wild ways to enter data. It will accept fractions. But for formula input you have to enclose them in quotes. They are not willing to offer what I am suggesting.
Computer designed entry, would allow entry of data with no parenthesis. Pretty much ever.
I enter all kinds of data all the time. I draw a lot of things that are built exactly like they are seen and measured in Cadd. I don't just draw to make a pretty picture. I draw to save wasted work and time.
Those pipes are cut and marked for the bender using the cadd input, bent up and they fit exactly as you see them there. I can draw up a rail now, entering in the stoop data as well, in under five minutes. And have fun with it.
http://www.rockwelder.com/EastWood/mathrail/mathrail.html
Sincerely,
William McCormick
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| William McCormick |
| Posted: Thu Apr 10, 2008 7:47 pm Post subject: Re: Out Of Order! |
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| serpicojr wrote: |
| William McCormick wrote: |
| I learned math in a place that at the time was considered the math capital in the World. |
William, really, Freeport, NY, was never the math capital of the world.
And, William, you've got to get off of this idea that math is somehow hampered by parentheses. It's not. |
Parenthesis, unnecessary work, plus unnecessary input, plus extra sorting, Parenthesis, times wasted time, equals error.
I just do not normally talk or think like that. But you inspire me to it. Ha-ha.
You do not like the idea of probably never using parenthesis for entering formulas?
Sincerely,
William McCormick
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| MagiMaster |
| Posted: Thu Apr 10, 2008 8:02 pm Post subject: |
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Question: How would you express (1+2+3+4)*(5+6+7+8 )*(9+10) then?
Actually, if you don't like parenthesis, use prefix or postfix notation. No parenthesis required. |
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| William McCormick |
| Posted: Thu Apr 10, 2008 8:55 pm Post subject: |
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| MagiMaster wrote: |
Question: How would you express (1+2+3+4)*(5+6+7+8 )*(9+10) then?
Actually, if you don't like parenthesis, use prefix or postfix notation. No parenthesis required. |
1+2+3+4X5+6+7+8X9+10
What is neat about this is that you can just type in the numbers first. And then choose your weapon.
Sincerely,
William McCormick
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| MagiMaster |
| Posted: Fri Apr 11, 2008 1:22 am Post subject: |
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| So how do you type in 1+2+3+(4*5)+6+7+(8*9)+10? I still think you should look at prefix notation. |
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| thyristor |
| Posted: Fri Apr 11, 2008 6:04 am Post subject: |
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I don't really understand what you mean in your example 10+11+13+14 1/2X5/4. It could easily be misinterpreted. How do you know that you don't mean 10+11+13+14+5/2/4?
And then you say that when you talk you don't say paranthesis, but then maybe you've noticed that this can cause problem.
If I say take the sqrt of 2 times 10 you don't know if I mean 10*(sqrt of 2) or sqrt of 20.
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| Harold14370 |
| Posted: Fri Apr 11, 2008 6:15 am Post subject: |
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| It reminds me of the calculators with reverse polish notation. They had their following, but most of us don't spend our days inputting numbers in a calculator. So we didn't want to learn a new system and just bought the algebraic kind. |
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| serpicojr |
| Posted: Fri Apr 11, 2008 8:50 am Post subject: |
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| Yeah, William, you should check out Polish/prefix of reverse Polish/postfix notation. It requires fewer symbols than your notation and is completely parenthesis-free. |
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| William McCormick |
| Posted: Fri Apr 11, 2008 4:47 pm Post subject: |
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| thyristor wrote: |
I don't really understand what you mean in your example 10+11+13+14 1/2X5/4. It could easily be misinterpreted. How do you know that you don't mean 10+11+13+14+5/2/4?
And then you say that when you talk you don't say paranthesis, but then maybe you've noticed that this can cause problem.
If I say take the sqrt of 2 times 10 you don't know if I mean 10*(sqrt of 2) or sqrt of 20. |
In the beginning I explained that the "/" symbol or fraction/ratio symbol, along with the original division symbol "÷". Meant two different things. They really still do, but no one has ever fought one way or the other. I would have but, I don't really know anyone that really likes math. Most just want to use it to get a job or play some games with it, or look smart.
The division symbol "÷" means to take everything to the left of the symbol in the formula, and divide it by everything to the right of it in the formula.
The fraction symbol "/" meant to just take the number to the left and the number to the right and turn it into a ratio, an entity. That would get parsed first by the computer or even a human parsing the equation.
The small "x" compared to the capital "X" also meant two different things. The small x turned two things being multiplied into a separate entity, that also got parsed first, much like a fraction but instead using multiplication.
The large "X" took everything on the left in the formula and multiplied it by everything on the right. If you had a situation where there were, a bunch of big "X"s you would just take them as they came. And multiply everything that had already been tallied up to that point, on the left and multiply it by everything to the right up to the next large "X" and so on and so on. Also if you hit a division symbol it signified a split in the formula. It was beyond anything we could ever hope for now with our current system.
To be honest most common formulas are not that long. If only for ease of manipulation. But you can make really nice to understand formulas with this system. Just like you would say them to someone in a conversation.
Being able to think and enter without first deciding where to put parenthesis, is a Godsend. I know I enter measurements by the hundreds into the computer. I am forever, using the best system available, but still needing to use parenthesis.
I used to use this method in school. It became very easy, but we had no real things to solve with it. I wanted to be out building a space ship. Not practicing math.
No more parenthesis, just talk out the equation or measurements and out pops the answer or object. It would be remarkable to see done on a computer.
Sincerely,
William McCormick
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| serpicojr |
| Posted: Fri Apr 11, 2008 6:51 pm Post subject: |
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So William... the number of ways of putting n-1 pairs of parentheses around n numbers (and hence the number of ways of parsing n-1 applications of a binary operation, e.g. division, to n numbers) is given by the (n-1)st Catalan number (Sloane's sequence A000108), which is:
C(n-1) = (2(n-1))!/n!(n-1)!
The number of ways of writing n-1 divisions between n numbers, given your notation, in 2n-1. C(n-1) > 2n-1 for n > 3:
C(n-1) = (2(n-1))!/n!(n-1)! = (2n-2)(2n-3)...(n+1)/(n-1)(n-2)...2*1
Pairing numbers on top with numbers on bottom, we have:
(2n-2)/(n-1), (2n-3)/(n-2), ..., (n-1)/2
(2n-2)/(n-1) = 2, and each of the remaining n-2 numbers is strictly greater than 2, so their product is greater than 2n-1. Thus your system cannot be parenthesis-free--there will be ambiguity for long enough expressions in your system. |
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| MagiMaster |
| Posted: Fri Apr 11, 2008 7:51 pm Post subject: |
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Seriously. If you hate parenthesis so much, you'd love Polish notation. It was designed partly to eliminate them.
For example,
Normal: (1+2+3+4)*(5+6+7+8 )
Yours: 1+2+3+4X5+6+7+8
Polish:* + + + 1 2 3 4 + + + 5 6 7 8
And,
Normal: 1+2+3+(4*5)+6+7+8
Yours: 1+2+3+4x5+6+7+8
Polish: + + + + + + 1 2 3 * 4 5 6 7 8
What you're talking about is called order of operations. It's what parenthesis are designed to overcome. |
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| William McCormick |
| Posted: Fri Apr 11, 2008 9:02 pm Post subject: |
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| serpicojr wrote: |
So William... the number of ways of putting n-1 pairs of parentheses around n numbers (and hence the number of ways of parsing n-1 applications of a binary operation, e.g. division, to n numbers) is given by the (n-1)st Catalan number (Sloane's sequence A000108), which is:
C(n-1) = (2(n-1))!/n!(n-1)!
The number of ways of writing n-1 divisions between n numbers, given your notation, in 2n-1. C(n-1) > 2n-1 for n > 3:
C(n-1) = (2(n-1))!/n!(n-1)! = (2n-2)(2n-3)...(n+1)/(n-1)(n-2)...2*1
Pairing numbers on top with numbers on bottom, we have:
(2n-2)/(n-1), (2n-3)/(n-2), ..., (n-1)/2
(2n-2)/(n-1) = 2, and each of the remaining n-2 numbers is strictly greater than 2, so their product is greater than 2n-1. Thus your system cannot be parenthesis-free--there will be ambiguity for long enough expressions in your system. |
Did you leave out the order, that is involved? You can have two in line division symbols "÷" in the same formula. I cannot see any end to it at all.
Sincerely,
William McCormick
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