Regarding mathematical operators

[tahayassen]

One basic operator is addition. In order to add the any number x number of times, multiplication was invented. In order to multiply any number x number of times, exponentiation was invented. What if we want to raise a number to a power x number of times? How come we didn't invent that?

Also, why is it that addition and multiplication use symbols, but exponents are just simply super-scripted?

Why do we have the opposite operations of addition and multiplication but there is no opposite operation of exponentiation?

Edit: I realize now that the opposite operation of exponentiation is the radical.

 

[symbolipoint]

This in not correct:

Why do we have the opposite operations of addition and multiplication but there is no opposite operation of exponentiation?

The opposite of the addition operation is subtraction, but we would call it the inverse operation. The opposite of multiplication operation is division, but we call it the inverse operation. This will become clearer as you study.

 

[tahayassen]

This in not correct:



The opposite of the addition operation is subtraction, but we would call it the inverse operation. The opposite of multiplication operation is division, but we call it the inverse operation. This will become clearer as you study.

So if they are opposites and not inverses, then why do we call them inverse operations?

 

[tahayassen]

Also, why is it that AA^-1 = I i.e. the inverse of a matrix behaves like a reciprocal (multiplicative inverse) in regular algebra? Shouldn't it be called the reciprocal or at the very least, the multiplicative inverse?

 

[Mandelbroth]

One basic operator is addition. In order to add the any number x number of times, multiplication was invented. In order to multiply any number x number of times, exponentiation was invented. What if we want to raise a number to a power x number of times? How come we didn't invent that?

"We" did invent it. Tetration is repeated exponentiation, pentation is repeated tetration, and so on. Hyperoperations (Look them up).

Also, why is it that addition and multiplication use symbols, but exponents are just simply super-scripted?

Why do you name your children Michael or Robert? In truth, it comes down to how really smart old dead people wanted to notate it.

Why do we have the opposite operations of addition and multiplication but there is no opposite operation of exponentiation?

Your use of the term "opposite operation" is painful to look at. You could mean SO MANY things with that. For example, a radical could denote taking the nth root of a number, and could be considered an "opposite operation". Logarithms could just as well apply as an "opposite operation" on exponentiation. Communication is important.

 

[tahayassen]

Thank you for answering my questions Mandelbroth. I think you may have missed post #4 though.

Also, a little bummed out that hyper-operations already exist since I thought I was onto something new. I got really excited and even wrote my own notation.

 

[symbolipoint]

Thank you for answering my questions Mandelbroth. I think you may have missed post #4 though.

Also, a little bummed out that hyper-operations already exist since I thought I was onto something new. I got really excited and even wrote my own notation.

Also, why is it that AA^-1 = I i.e. the inverse of a matrix behaves like a reciprocal (multiplicative inverse) in regular algebra? Shouldn't it be called the reciprocal or at the very least, the multiplicative inverse?

Are you studying a specific course?

 

[tahayassen]

Are you studying a specific course?

I just finished year 1 linear algebra.

 

[Number Nine]

Also, why is it that AA^-1 = I i.e. the inverse of a matrix behaves like a reciprocal (multiplicative inverse) in regular algebra? Shouldn't it be called the reciprocal or at the very least, the multiplicative inverse?

It's called the inverse of A because multiplication by A gives the identity, which is the definition of inverse. "Multiplicative inverse" is redundant in almost all cases.

 

[micromass]

It's called the inverse of A because multiplication by A gives the identity, which is the definition of inverse. "Multiplicative inverse" is redundant in almost all cases.

Well, you also have additive inverse of a matrix :tongue2:

I think the OP his questions would very likely be cleared up by an abstract algebra course.

 

[tahayassen]

So as micromass said, there is the additive inverse and the multiplicative inverse of a matrix. So why is it that when we say inverse of a matrix, we are referring to the multiplicative inverse? Shouldn't it be called the reciprocal? I think it's misleading.

 

[pwsnafu]

So as micromass said, there is the additive inverse and the multiplicative inverse of a matrix.

Every matrix has an additive inverse, it's the matrix with every entry negated.
Not all matrices have multiplicative inverses. So when we use "inverse" we mean multiplicative because it's significant.

So why is it that when we say inverse of a matrix, we are referring to the multiplicative inverse? Shouldn't it be called the reciprocal? I think it's misleading.

Reciprocal refers only to fractions (or "things that behave like fractions", i.e. fields). That is
$\left(\frac{a}{b}\right)^{-1} = \frac{b}{a}$. In fields every non-zero element has a multiplicative inverse, so we don't use use the term "inverse" as it isn't significant.

Later in your studies you'll come across rings, which is where this nomenclature originates from. It's make more sense then.

 

[symbolipoint]

So as micromass said, there is the additive inverse and the multiplicative inverse of a matrix. So why is it that when we say inverse of a matrix, we are referring to the multiplicative inverse? Shouldn't it be called the reciprocal? I think it's misleading.

A matrix is a different thing than a Real Number. A matrix can have a multiplicative inverse. Context (meaning the situation in which the matrix is of interest) for any matrix may tend to support it being either the additive inverse or the multiplicative inverse.

 

[tahayassen]

Makes sense.

 

[symbolipoint]

A matrix is a different thing than a Real Number. A matrix can have a multiplicative inverse. Context (meaning the situation in which the matrix is of interest) for any matrix may tend to support it being either the additive inverse or the multiplicative inverse.

Maybe making clearer be a good idea...

Take any real number other than zero. Call it a. It has an inverse (meaning here multiplicative inverse) a^-1. This allows the equation, aa^-1=1. Also, a^-1a=1. Reciprocal is a multiplicative inverse of a Real Number. That is for Real Numbers.

Matrices are not always like that. Take any square matrix. MAYBE it has a multiplicative inverse and maybe it does not. What if you have a square matrix, A. Then it might or might not have an inverse, A^-1. If it does, then AA^-1=I, and A^-1A=I. Sometimes, there is a matrix B for which AB=I, but BA=\=I; or that BA=I but AB=\=I. For matrix multiplication, AB and BA are not always the same.

 

[micromass]

Sometimes, there is a matrix B for which AB=I, but BA=\=I; or that BA=I but AB=\=I.

Actually, it might be surprising that this is not the case! If you can find a matrix B such that AB=I, then that actually implies that BA=I. This is highly nontrivial but extremely useful.

 

[symbolipoint]

Actually, it might be surprising that this is not the case! If you can find a matrix B such that AB=I, then that actually implies that BA=I. This is highly nontrivial but extremely useful.

Maybe I misunderstood something. BA is not always equal to AB, but I made my comment in regard to product being the identity matrix I. Plenty enough for me to both learn and relearn about Linear Algebra.

 

[micromass]

Maybe I misunderstood something. BA is not always equal to AB, but I made my comment in regard to product being the identity matrix I. Plenty enough for me to both learn and relearn about Linear Algebra.

Yea, it's true that $AB\neq BA$ in general. But if $AB=I$, then it can be proven that $BA=AB=I$. This follows essentially from the rank-nullity theorem.