Here's something I've been doing with numbers lately

(1,2,3,4,5)

|----------- |    
(1,2,3,4,5)    
(Note dotted line should be at very top of the two lines connecting 1 to 5)

5+1 = 6
2+4 =6  
3+3 = 6

Now if you were repeat this process adding a longer line from 2 to 4 and draw a straight line at the top of the horizontal lines connecting 2 to 4, you'll start creating a pyramid. A better illustration would be a larger set of numbers. For example 1 - 10 :

1 + 10 = 11
2+ 9 + 11
3 + 8 = 11
4 + 7 = 11
5+ 6 + 11
6+ 5 = 11
7 + 4 = 11
8 + 3 = 11
9 + 2 = 11
10 + 1 = 11

So my question is, have I stumbled on some mathematical formula that I have no idea of I'm investigating ? Or is this something trivial? I think this has something to do with "Power to the" type of formula. Like :

1 to the power of 1 = 1, 2 to the power of 2 = 4,  4 to the power of 4 = 8, 8 to the power of 8 = 16, and so on and so on....  

I may not fully understand algebra, but I do remember "to the power of's" .......

You are doing well, this is a good way to get your head inside the music of numbers. Play around with them and see the patterns pop out at you.

And you HAVE stumbled upon something. You are not the first, however.

A famous mathematician stumbled across this at school, many years ago, when his class was asked to add all the numbers from 1 to 100 together. He put his hand up much more quickly than the teacher expected and gave the correct answer. He had figured out a method of adding those numbers together, that being 100+1. 99+2 and so on. The correct answer, of course, being 50 * 101 = 5050.

What he had done was to come up with a clever way of adding together a sequence of numbers, which is extremely useful for probability calculations.

http://betterexplained.com/articles/tech...-1-to-100/

I googled it. The mathematician's name was Gauss.

The website in the link has some really interesting insights and discussion.

Another thing you can do is to look for how you can add up to get numbers that are easier to work with (like 10). Example:

1+9=10
2+8=10
3+7=10
4+6=10

That's four tens, or 40. But you're not done yet. You still have a 10 and a 5, adding up to 55 (40+10=50, 50+5=55).

I think playing around with numbers is a lot more satisfying than reading about numbers in those text books I found. It can get exhausting for me doing basic Addition and Multiplication training on my Nintendo DS emulator with the game Brain Age. Not to mention being very tiresome learning about Algebra. Here's an example of what's confusing me from the Wikipedia article on PreAlgebra :

1.   Solve:   x + 79 = 194

    Solution:  
    x + 79 = 194
    x + 79 - 79 = 194 - 79   (79-79 is 0, how the hell do you get equal 194 from ?)
    x = 115     (This part I understand)  


- 194
    79
ignore the 100's place 9-7 = 2 but we borrow from 7 cause you can't take 9 from four so you add ten. thus 14 -9 = 5, your answer, 115. (My ways of figuring it may be unorthodox ed).

Unless you subtract from both sides even if one side equals zero
  x + 79 - 79 = 194 - 79    
then subtracting 194 -79 would solve for x
thus  

115 + 79 = 194

But something comes along and confuses me even more:

Solution:  
    x - 56 = 604 (Where did we get 604 from?)
    x - 56 + 56 = 604 + 56 (So we add 56 by itself?)
    x = 660

So is the purpose of PreAlgebra is to get the student thinking in different ways to solve something, thus, thinking out of the box ?

That's the basic idea. For purposes of learning the math, it doesn't really matter where 194 came from. The question is what number can you add to 79 to get 194.

Maybe this concept might help. Subtracting is the same as adding a negative number.

x - 56 = 604
is the same as
x + (-56) = 604
From here, the method is the same as above.
x + (-56) - (-56) = 604 - (-56)
x = 660

How come my brain now feels worn out ? Is that even possible, has anyone else ever felt like your mind is literally pooped out ?

I keep thinking to add -56 within it's self cause of x. But it's a negative number and I though if you added -56 with it's self it'll equal 0 . And 56 + 56 = 112... I don't mean to be so frustrating, but there's something I'm not grasping here

You can look at it like this:

x + 79 = 194

The FIRST thing you should do when you see an equation is pick out the values that you know, and the values that are unknown.
x is unknown (you dont know how many x is).
194 is known, because 194 clearly equals 194.
79 is also known, because 79 clearly equals 79.

The SECOND thing you should do is try to put the known values on the SAME SIDE of the equation. The two sides of an equation are separated by the equal sign.

Look at the equation again: x + 79 = 194
The known value 79 is on the other side of the equation, but we want both 79 and 194 on the same side.
To do this, we have to set up a trade. Essentially you are going to steal 79 from the left side (and by stealing, you subtract), and add 79 to the right side.

This is how you steal a positive number:
79 - 79 = 0

So you do that to the equation; steal 79 and, then add it with 194.

-------

The other problem is the same thing, EXCEPT you are going to steal a negative number (-56). In order to steal (or cancel out) a negative number, you add a positive number.

To better understand positive and negative numbers, think of a thermometer. Lets say it is 79 degrees outside. In order to bring the temperature down to 0 degrees, you need to subtract 79 degrees of heat.
79 degrees - 79 degrees = 0 degrees.

Now, if it is -56 degrees outside (COLD!), and you want to raise the temperature to 0 degrees, then what do you do?
The answer is, you have to add 56 degrees.
-56 degrees + 56 degrees = 0

Therefore, in the second problem, to steal -56, you add 56. Once you have stolen the 56, you then add it to the other side of the equation.

hope that helps...

I though you subtracted 79 from 194 to find the value of x, cause isn't it like doing addition in reversed ? Or a different, alternative way ?

That's correct. I think SG must've misspoken. The rule is that whenever you add/subtract/multiply/divide on one side of the = sign, you have to do the exact same operation with the exact same number on the other side.

I remember this, from those times I was briefly in Algebra class. It doesn't matter if sg1008 mispoken, I appreciate any and all help all the same

Funny how it's all coming back to me

oops my bad, yeah i misspoke and misthought it out.

damn sleep deprivation...im going to make a new reply, sorry.

[I AM SO SORRY I COMPLETELY SCREWED UP THE FIRST TIME. IN MY HEAD I CAN DO IT FINE, BUT SOMEHOW I CONFUSED MYSELF IN THE EXPLANATION]

HERE IS A CORRECT VERSION:
You can look at it like this:

x + 79 = 194

The FIRST thing you should do when you see an equation is pick out the values that you know, and the values that are unknown.
x is unknown (you dont know how many x is).
194 is known, because 194 clearly equals 194.
79 is also known, because 79 clearly equals 79.

The SECOND thing you should do is try to ISOLATE the UNKNOWN value (x) on one side of the equation. The two sides of an equation are separated by the equal sign.

Look at the equation again: x + 79 = 194
The unknown is x, but it is coupled with a known value 79. In order to isolate x, we have to deal with 79.
To do this, set up REMOVAL. Essentially you are going to remove the known value 79 from the left side (and by removing, you subtract). BUT whatever you do to one side of the equation, you MUST do the same to the other side. So we also have to remove 79 from the right side.

This is how you remove a positive number:
79 - 79 = 0

So you do that to the equation; remove 79 from x by subtracting it, and then balance the equation by also subtracting 79 from 194.

-------

The other problem is the same thing, EXCEPT you are going to remove a negative number (-56). In order to remove (or cancel out) a negative number, you add a positive number.

To better understand positive and negative numbers, think of a thermometer. Lets say it is 79 degrees outside. In order to bring the temperature down to 0 degrees, you need to subtract 79 degrees of heat.
79 degrees - 79 degrees = 0 degrees.

Now, if it is -56 degrees outside (COLD!), and you want to raise the temperature to 0 degrees, then what do you do?
The answer is, you have to add 56 degrees.
-56 degrees + 56 degrees = 0

Therefore, in the second problem, to remove -56, you add a positive 56. Once you have added 56 to one side, you then add it to the other side of the equation to balance it out.

hope that helps...

This post was last modified: 07-02-2012 08:50 AM by sg1008.

It's okay sg1008 :/ Nobodies perfect, I was just saying I appreciate any help anyone could offer me. I've had days of sleep deprivation and I know how that's like, I totally screwed up my Debian partition  where I only have 10 GB's to install programs and 109 for everything else when I was totally sleep deprived. I've been having up-all-nighters since I was 9 yrs old.

I appreciate the fact that you would go out of your to explain something to me Thank you

It's okay sg1008 :/ Nobodies perfect, I was just saying I appreciate any help anyone could offer me. I've had days of sleep deprivation and I know how that's like, I totally screwed up my Debian partition  where I only have 10 GB's to install programs and 109 for everything else when I was totally sleep deprived. I've been having up-all-nighters since I was 9 yrs old.

I appreciate the fact that you would go out of your to explain something to me Thank you

You are most welcome! I like helping when I can. I am going to work on math tomorrow (ive decided Mondays are math days) so if I make any interesting observations I'll make a thread about it.