Transposition

[Yuseph]

Hey guys,

Ok only two issues and im finished with the transposition chapter.

Tell me why on earth would the result not be x = z - y ?

On the next picture i dont understand why 2pi squared has to become 4pi squared ?
I mean 2pi is just one entity it should be treated as 6,28^2. Not as 2^2 * pi^2. That one really got me lost.

 

[R.M.]

Here's an example: 3² = 5² - 4², but if your theory is correct, then taking the square root of each side would yield 3 = 5 - 4 --> 3 = 1.
A square root is the inverse of a 2nd power, NOT the inverse of a difference of squares. Note the order of operations here: the subtraction is occurring before the taking of the root, so therefore we cannot just simply cancel those powers.

For your second problem, your are squaring a fraction, which means you are multiplying the fraction by an exact copy of itself. Because you are multiplying 2 [MATH]\pi[/MATH] by 2[MATH]\pi[/MATH], you will get 4[MATH]\pi^2[/MATH]

 

[Yuseph]

I understand what you're saying but it s really weird that it d work with multiplication and division and not with substraction and addition. What confused me the most is its in total contradiction with the PEMDAS rule. I replaced by 5 3 and 4 too and of course if you put logic into it you cant go wrong but its just weird

 

[Dr.Peterson]

I understand what you're saying but it s really weird that it d work with multiplication and division and not with substraction and addition. What confused me the most is its in total contradiction with the PEMDAS rule. I replaced by 5 3 and 4 too and of course if you put logic into it you cant go wrong but its just weird

Actually, it's just as I'd expect.

We say that multiplication distributes over addition, a(b+c) = ab + ac; and exponentiation distributes over multiplication, (a*b)^c = a^c * b^c; but we wouldn't expect that exponentiation would also distribute over addition, (a+b)^c = a^c + b^c. Each has its own place.

And the PEMDAS "rule" is just a statement of how we interpret expressions; how numbers themselves work can't contradict that, any more than a statement about science could contradict the grammar of the language we are writing in. PEMDAS just says that [MATH]\sqrt{x^2 + y^2}[/MATH] means to square, then add, then take a square root; we can't do the square root before the addition unless some property of numbers tells us that those two different expressions have the same value -- and they don't.

However, it happens (ultimately, I think, for good reasons) that they fit together nicely: PEMDAS tells us that Exponentiation is more powerful in some sense than Multiplication (and Division), and Multiplication (and Division) is more powerful than Addition (and Subtraction) -- and those are exactly the relationships in which operations distribute!