This probably seems like a really stupid question. I am no math expert; I am in an Algebra II high school class. Here it goes:
addition can be interchanged such that
a + b = c
b + a = c
but why not subtraction?
a - b = c
b - a does not equal c
I don't quite understand this
and with division and multiplication i thought that:
a*b/c = d
and a*(b/c) = d
I have always been told division and subtraction are interchangeable in school. I just looked at this problem and realized Ive been fooled:
6/3*2 = 4
6/(3*2) = 1
I am so confused. Can someone just clear up this whole order of operations mumbo jumbo for me. I would like to know exactly what division and multiplication are and what arithmetic works in relation with one another. I am in Algebra II and I can't believe I don't understand this.
Best Answer
You may notice that subtraction can be seen as a special case of addition, i.e. $$a-b=a+(-b)$$ Thus by commutative law, $$a-b=a+(-b)=(-b)+a=-b+a$$ It is certainly not $b-a$.
Similarly, division is as a special case of multiplication, i.e. $$\frac{a}{b}=a\cdot(\frac{1}{b})$$ Where $\frac{1}{b}$ means the inverse element of $b$ with respect to multiplication.
Therefore, $$\frac{a}{b}\cdot{c}=a\cdot\frac{1}{b}\cdot{c}$$ However, $$\frac{a}{b\cdot{c}}=a\cdot\frac{1}{b\cdot{c}}=a\cdot\frac{1}{b}\cdot\frac{1}{c}$$ And they are certainly not equal.
Edited: Since you are motivated to pursue advanced math, I have a little idea to share.
I assume you have intuitively understood the arithmetic about nature numbers. It is based on this simple idea: the one-to-one correspondence between (abstract) nature numbers and (concrete) things. Then you easy find that:
$(+, 1)$ The commutative law of addition $$a+b=b+a$$
$(+, 2)$ The associative law of addition $$a+(b+c)=(a+b)+c$$
$(\times, 1)$ The commutative law of multiplication $$a\cdot{b}=b\cdot{a}$$
$(\times, 2)$ The associative law of multiplication $$a\cdot(b\cdot{c})=(a\cdot{b})\cdot{c}$$
$(\times, 3)$ The identity element of multiplication, i.e. $1$ $$a \cdot 1=1 \cdot a=a$$
$(+, \times)$ the distribution law of multiplication to addition $$a \cdot (b+c)=a\cdot {b}+a \cdot {c}$$
A possible intuitive approach has been shown by Paul Sinclair. And you also have noticed that addition and multiplication can be defined as recursive operation. That is, $$a+b:=a+\underbrace{1+1+1+...+1}_{b \text(terms)}$$ And $$a\cdot{b}:=\underbrace{a\cdot{a}...\cdot{a}}_{b \text(terms)}$$
You also have a nature idea to introduce the inverse operation, i.e. substitution and division. But you may find that $3-5$ is illegal in this stage (it has no definition), and neither is $\frac{3}{5}$.
But that is not hard for you, just noticed that you can define $\frac{3}{5}$ as a ratio: There certainly can be a thing $k$ such that $k\cdot{5}=3$, then we can denote $k$ by $\frac{3}{5}$. Here come the rational numbers! And you may just find
$(\times, 4)$ The reverse element of multiplication $$\text{For every number $a$ there is a number $b$ such that $a\cdot{b}=b\cdot{a}$}$$
And we can easily find that by definition, $b=\frac{1}{a}$. (the "number" here means nature number and positive rational number.)
However you are not satisfied. Sometimes you need to find a way to denote "nothing", so you need
$(+, 3)$ The identity element of addition, i.e. $0$ $$\text{For all $a$, $a+0=0+a=a$}$$
As far as I'm concerned, historically, in a relative long time, $0$ or similar notations were initially but for this kind of convenience.
A similar idea for convenience is negative number, it was originally used to express debt. With introducing negative numbers, we can derive:
$(+, 4)$ The reverse element of addition $$\text{For every number $a$ there is a number $b$ such that $a+b=b+a=0$}$$
And we can also easily find that by definition, $b=-a$.
Indeed, It took people a lot of time to accept the philosophical aspect of "nothingness", or to understand how debt times debt will be income (this was not philosophical confusion initially, but a result of wrong metaphor; however, it finally became one, but that is another story.)
But for mathematics, the most horrible thing is division by 0, it is certainly not legal. There is no help for it, so we have to ban it. Indeed, this is the only difference between addition and multiplication in abstract sense. If you consider about a more general case such as real numbers, the original, recursive idea will failed, or, at least, no more intuitive. Thus you have to define addition and multiplication in a more abstract way, that is, introducing the axioms of addition and multiplication, i.e. $(+,1)-(+,4),(\times,1)-{(\times,4)},(+,\times)$ I've mentioned before. If you compare $(+,1)-(+,4)$ with $(\times,1)-(\times,4)$, you will find they are nearly the same.
A set equipped with two operation satisfying those axioms will be the field which trb456 mentioned. (To say it clearly, the result of those operations must always within that set, and the identity element of addition can't be the same with that of multiplication.) We need this term for other reason: because there are a lot of other structures satisfying this definition. And modern algebra, though I am totally not familiar with it yet, seems like a study of mathematical structures.
This story is obviously not the real history of numbers and their arithmetic, and I just want to introduce a half-intuitive approach to understand those things. I omitted a lot of details, or maybe I just got things wrong. But I hope you enjoy it.