Excellent wandering through the dark woods and bumping into trees. Why not step out into the light of day?
"=" is a comparison symbol. It doesn't force things. It will help you remember equivalence, it will not create it.
I've always used equations, but some time ago i have started to ask myself "why the heck can i pick 2 things, put a = between them, and get to conclusions?".
I've thought about the following, please, correct me if i am wrong. Tell me if this is a correct explanation for equations.
I say X+1 = 5
when is this true?
exactly when X+1 is 5
and i think "okay....but how does that help?" it could, but i dont want to know how much X+1 needs to be, i just want to know how much is X
then i think
if i say X+1 = 5, then i can remove 1 from both sides and will get an equivalent sentence.
I do it
X+1-1=5-1
i get X=4
what does that mean?
X+1 = 5 is true if X=4
Tell me if i am correct up to this point.
Now i thought about another theory
X+1=X
i apply me previous idea, and i get
0=1
what does that mean?
it means that X+1 would equal X if in my universe 0 was the same as 1
well, i could create my own universe in which 0 equals 1, but i want to me in the numbers universe, so i wont accept that solution, and so there isn't any other.
Did i do anything wrong in my reasoning? Please tell me, i'm no mathematician, and this problem is killing my brain, i need to explain equations.
Thanks in advance
Yep you have fallen off track as pointed out by TKHunny. I think your choice of symbol is important here.
For x+1 = x , the left hand side x is assumed to be the same value as the right hand side x, if they are the same number they can never have a difference of 1.
You can say x+1= y if you like.
I appreciate your ability to try to think against the heard, and ask "Why?"
To think about it another way for your examples is to imagine them in an xy graph. When will y=x+1 intersect line y=5? It will at x=4. When will y=x+1 intersect the line y=x? It leads to a contradiction in legally solving it, therefore it was erroneously assumed to be true in the first place. Kind of like setting up a proof by induction.
At least, that's my two cents.
I tried to invent different variations of "=" while I was in college. I used various over-letters to provide additional intent.
M ==> Must Be Equal To (usually the end of a proof)
B ==> Better Be Equal To (usually the beginning of a proof)
There were a few others, but they don't come to mind right at the moment.
A wise professor put up with this for a little while, but finally handed back a paper with the note, "Please don't invent new notation. There is already way too much."
"+/-" may be the most over-defined. See if you can find all five (5) meanings. Don't forget the inverted one.
You must NOT believe that every symbol means the same thing in every context. You must think about what it means in the context where it is encountered.
My views. I welcome others'.
Haha XD, i hope so.
But tell me something, is it right to say that an equation begins with an assumption?
I assume X+1 equals 5, so i can get to other conclusions from that assumption, such as that X equals 4
which means,
if X+1 is said to be equal to 5, then X equals 4