Dialogue de sourds about mathematics: Are exponents substractions in disguise?

[Rystil Arden]

Eh?

Prove it.

Given:

That doesn't prove anything. You still didn't get rid of the fractional exponent until you calculated the exponent directly (without multiplication) in the final step. You never transformed it into a series of mutliplications (and its impossible to do so).

 

[Patryn of Elvenshae]

That doesn't prove anything.

Nor does this.

You just said that "fractional exponents prove that not all exponents can be rewritten as mulitiplication."

Prove it.

 

[Rystil Arden]

Nor does this.

You just said that "fractional exponents prove that not all exponents can be rewritten as mulitiplication."

Prove it.

I can prove it in the following simple manner (by presenting one instance of an impossible case):

Expand X^(1/2) into only multiplication (no exponents or square root symbols allowed).

 

[Altamont Ravenard]

Expand X^(1/2) into only multiplication (no exponents or square root symbols allowed).

If there are any mathematical historians, I have a question relating to what would be Rystil's proof:

Someone, one day, decided that x * x = x²
Were roots "invented" at that time, or was it later that it was decided that sqrt(x) = x^½?

(it probably doesn't matter )

(btw, x^-1 = 1/x)

AR

 

[Deset Gled]

If there are any mathematical historians, I have a question relating to what would be Rystil's proof:

Someone, one day, decided that x * x = x²
Were roots "invented" at that time, or was it later that it was decided that sqrt(x) = x^½?

(it probably doesn't matter )

(btw, x^-1 = 1/x)

AR

A quick look at some of my reference books leads me to believe the first recorded text involving exponents was the Indian text "Anoyogdwar Sutra" (from sometime around 500 CE). It includes both exponents and roots, and the relationships between them. I don't know if the authors knew the concept of irrational numbers, but they did understand infinty.

And for the record, the idea that an exponent is the same thing as multiplication is a gross simplification. That's how it's often taught to students first encountering it in a discrete case, but it quickly falls apart when higher mathematics are brought into play. IIRC, the modern definition of an exponent is derived from the logarithmic function (it's the only way to account for irrational powers while maintaining continuity). For more information, feel free to consult any introductory calculus book at your local library.

Why was the definition of an exponent important?

 

[Rystil Arden]

A quick look at some of my reference books leads me to believe the first recorded text involving exponents was the Indian text "Anoyogdwar Sutra" (from sometime around 500 CE). It includes both exponents and roots, and the relationships between them. I don't know if the authors knew the concept of irrational numbers, but they did understand infinty.

And for the record, the idea that an exponent is the same thing as multiplication is a gross simplification. That's how it's often taught to students first encountering it in a discrete case, but it quickly falls apart when higher mathematics are brought into play. IIRC, the modern definition of an exponent is derived from the logarithmic function (it's the only way to account for irrational powers while maintaining continuity). For more information, feel free to consult any introductory calculus book at your local library.

Why was the definition of an exponent important?

Why was the definition of exponent important? Patryn claims they are indistinct from multiplying and thus the exponent step in order of operations (PEMDAS) is irrelevant.

 

[Deset Gled]

Why was the definition of exponent important? Patryn claims they are indistinct from multiplying and thus the exponent step in order of operations (PEMDAS) is irrelevant.

Ah. I see. Well, that's fairly simple to rectify.

Consider: x*y^z

Are the following statements equivalent for all case of x,y, and z (assume the set of all real numbers, we can neglect imaginary numbers for simplicity)?

x*y^z
(x*y)^z
x*(y^z)

If exponents are the same as muliplication, then the Associative Axiom of Multiplication states they are all the same. If you can find a case where they're not, then either the axiom is wrong (which I seriously doubt), or an exponent is not the same as multiplication.

Trying to use the formal definition of an exponent to show this is kind of like using a chainsaw to cut butter.

 

[Rystil Arden]

Ah. I see. Well, that's fairly simple to rectify.

Consider: x*y^z

Are the following statements equivalent for all case of x,y, and z (assume the set of all real numbers, we can neglect imaginary numbers for simplicity)?

x*y^z
(x*y)^z
x*(y^z)

If exponents are the same as muliplication, then the Associative Axiom of Multiplication states they are all the same. If you can find a case where they're not, then either the axiom is wrong (which I seriously doubt), or an exponent is not the same as multiplication.

Trying to use the formal definition of an exponent to show this is kind of like using a chainsaw to cut butter.

Hahahahahahahaha. Read the older posts of this thread if you think it is that simple to rectify (like I once did). You'll get a kick out of it.

 

[apesamongus]

Your examples have nothing to do with what he is saying.

Actually, yes they do. Please stop being wrong. If the steps are done all at the same time, then you can arrange the order at will. In this case, you cannot.

 

[Hypersmurf]

Actually, yes they do. Please stop being wrong. If the steps are done all at the same time, then you can arrange the order at will. In this case, you cannot.

In BEDMAS, the order of operations, the E stands for 'exponent'.

In the expression 2y², I see the ² as representing the operation "raise to the power of 2".

Patryn sees the ² as representing the operation "multiply by y"... and since multiplication is commutative, order doesn't matter. It could read ²2y, as long as we understand ² to mean "multiply by y", without changing the result.

-Hyp.