Why is $x$ divided by $n$ the same as $1$ divided by $n$ times $x$?

Question

I'm very new here and my question must have been answered before, but I don't know how to search this site with problems like mine.

My question is why $x/n$ is the same as $1/n*x$?

Also, if you don't mind, tell me how to search this site effectively with questions like mine. When I type something like this, every imaginable question shows up because of the search of symbols $x$ and $n$ and so on I guess.

Thanks a lot!

Answer 1

Division with fractions is defined to be multiplication by a reciprocal.

For example, $$\frac23 \div \frac 25 = \frac23 \times \frac52 = \frac53.$$

When the divisor doesn't appear to have a denominator, remember it is implicitly $1$:

$$x \div n = \frac{x}1 \div \frac{n}1 = \frac{x}1\times\frac1{n} = \frac{x}{n}.$$

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Addendum: You may wonder why division with fractions is defined in this way.

It can be arrived at by regarding division as a "bigger" fraction:

$$\frac ab \div \frac cd = \frac{\;\frac ab\;}{\frac cd}$$ $$=\frac{\;\frac ab\;}{\frac cd}\times 1$$ $$=\frac{\;\frac ab\;}{\frac cd}\times \frac{\;\frac dc\;}{\frac dc}$$ $$=\frac{\frac ab \times \frac dc}{\frac cd \times \frac dc}$$ $$=\frac{\frac ab \times \frac dc}{1}$$ $$=\frac ab \times \frac dc$$

Answer 2

Several ways to think of it.

Perhaps the most practical is: $\frac xn$ usually means (depending on what context; how abstract; what your philosophy of what mathematics actual is; and what type of conversationalist you are at cocktail parties) you have a quantity of $x$ stuff and you are dividing it into $n$ equal pieces how many is each piece. Well, you have $x$ stuff and each $1$ of those $x$ stuff is split into $n$ pieces so each $1$ is split into $n$ pieces $\frac 1n$ in size each. And each of the $x$ stuff contriibute on $n$th of itself to the whole thing and you have $x$ of these $\frac 1n$ pieces. So that is $x\cdot \frac 1n$ of the stuff.

This implies $n$ is a whole number and although $x$ might not be it is somehow measurable in relation to quantities of $1$.

More abstract and mathematical:

Multiplication is a "binary operation" so that given any two real numbers $a$ and $b$ if you "do the operation on them" then $a\cdot b$ will equal some real number always. You can't have $3\cdot 7=21$ and $2\sqrt 3\cdot \sqrt 6=6\sqrt 2$ but $9\cdot 7$ doesn't equal anything at all, or $5\cdot 1=$ pink elephant. $a\cdot b$ always equals some number.

By axiom for any real number $a\ne 0$ there is a real number $a^{-1}$ called the multiplicative inverse, which has the property that $a\cdot a^{-1} =1$. For notation and notation only we write this number as $\frac 1a$. $\frac 1a$ is a real numbers so $x\cdot \frac 1n$ must equal some number. For notation purposes we write $x\cdot \frac 1n$ as $\frac xn$. It means the number: $x$ multiplied by the number that when multiplied by $n$ would give you $1$.

That is the definition there is no such thing as "division". The concept of $x\div n = \frac xn$ will mean in the context of Abstract Algebra way of thinking: Take the number $x$ and perform the "multiplication" operation on it with that number than if you did the operation with $n$ would give you $1$.

.....

So those are two very different but valid ways of thinking of what math is.

1) is practical: Look, take some pizzas and cut them into slices. Look it has to work because splitting a bunch of pizzas won't change just because of the order you do it in. (So just do it, kid. That's how the world works.)

2) is abstract and axiomatic: Mathematics is a game of definitions and axioms and rules. This is so because we chose to define the axioms so that it is so. (Now shut up and eat your soup.)

Answer 3

$\dfrac{x}{n}$ defines $x$ pieces of size $\dfrac{1}{n}$, the way you would call $\dfrac{3}{4}$ of an orange three quarters of an orange, or how a mark of $\dfrac{43}{50}$ in a test is 43 points worth, because one mark is defined as $\dfrac{1}{50}$ of the possible 50 marks available.