Basic division and multiplication

Question

We know that 2 x 2 /2 can be solved by removing '2' from the denominator and numerator, we can't do the same if the operation was addition, These "rules" have been established based on the understanding of these operations and how these interact with each other.

Up until now, I've just been accepting it as a rule, but this causes a lot of discomforts while doing math.

I generally get good grades in math, but that's only by following these rules and patterns but there is no genuine understanding as to why we can remove "2" from the numerator and denominator in 2 x 2 /2 but we cannot in 2 + 2/2.

Secondly, 2 x 2 / 2 can be solved by removing "2" from the numerator and the denominator and it gives the same result as 4/2, how?. Is it an axiom?

I hope to gain an intuition for the above concepts, so I can be free from my long-held guilt.

Thanks.

Answer 1

We have that

$$\frac{2\times 2}{2}=2\times 2\times \frac12$$

and by associative property, which is an axiom we have that

$$(2\times 2)\times \frac12=4\times \frac12=2$$

$$2\times \left(2\times \frac12\right)=2\times 1=2$$

are two equivalent way to simplify the expression.

Answer 2

When you remove the $2$ from the numerator and denominator what you are really doing is multiplying the fraction by $1$ in the form $$\dfrac {\dfrac{ \ 1\ }{\ 2\ }}{\dfrac{ \ 1\ }{\ 2\ }}$$ You can do the same with $2+\frac 22$, which becomes $2+\frac 11=3$. When you multiply $2$ by $\frac 12$ you get $1$, not nothing. In multiplication just removing it works because $1$ is the multiplicative identity, but it is a bad idea to think of it as removing. That will lead to errors like the one you cite.

Answer 3

$\dfrac {2 \times 2}{ 2 }$ can be solved by removing $2$ from the numerator and the denominator and it gives the same result as $\dfrac 4 2$, how? Is it an axiom?

No, it is not an axiom. But, as for every rule for operating with numbers, it is obviously justified by axioms.

Regarding the rationals (i.e. fractions) they are an ordered field.

Regarding division, the axioms state that :

If $a \ne 0$, then the equation $a \cdot x = b$ has a unique solution : $x = \dfrac b a = b \cdot a^{−1}$.

Thus, $a^{−1}$ is the multiplicative inverse of $a$.

We have :

$\dfrac {a \cdot a} a = (a \cdot a) \cdot a^{−1}$.

By Associativity of product we have that :

$(a \cdot a) \cdot a^{−1} = a \cdot (a \cdot a^{−1})$.

But $(a \cdot a^{−1})=1$ and again by axiom :

$a \cdot 1=a$.

Putting all together, we get :

$\dfrac {a \cdot a} a = a$.

---

At an "intuitive level" it is simply a metter of multipling and dividing quantities.

We multiply two by two the get four and then we divide four by two, getting two.

In real life applications of numerical oeprations we do not "cancel" anything.

Answer 4

I'm not quite sure how advanced in mathematics you are so this is going to be the real simple explanation.

The fraction bar is notation for division. $\frac{3}{4}$ represents the number that is defined as the result of the operation $ 3\div 4.$

Therefore $$ \frac{2}{2} = 1$$ because $2 \div 2 = 1.$ In your case

$$ 2 \times \frac{2}{2} = 2 \times 1 = 2$$ So it looks like we just "removed" the two's. This only works because we end up by multiplying by $1$ so it is in general best not to talk about it like that.

This obviously doesn't work for addition

$$ 2 + \frac{2}{2} = 2 + 1 = 3.$$

Also $$ \frac{4}{2} = 4 \div 2 = 2 \quad \text{ so } \quad 2 \times \frac{2}{2} = \frac{4}{2}.$$

Answer 5

Before 'nailing down' multiplication and division, the first step is be totally comfortable with one concept - 'adding up' positive numbers. As an intuitive map for images held in the brain, you can take your unit of measure to be '$1$ Apple Pie' or ''$1$ Meter'.

If you have '$1$ Apple Pie' you can certainly imagine having one in the left hand and one in the right hand and combining them. If we use the notation [AP] for this unit,

$\tag 1 1 \text{ [AP]} + 1 \text{ [AP]} = (1 + 1) \text{ [AP]} = 2 \text{ [AP]}$

Notice that we can now begin to understand multiplication with an integer as 'repeated addition' notation:

$\tag 2 2 \times 1 \text{ [AP]} = 1 \text{ [AP]} + 1 \text{ [AP]} = (2 \times 1)\text{ [AP]}$

Now what is, say, $\frac{1}{4} \text{ [AP]}$? This looks good

$\tag 3 4 \times \frac{1}{4} \text{ [AP]} = 1\text{[AP]}$

This fractional notation, once you think of what we are trying to represent, is a marvelous notation.

Addition is about combining quantities. So $2\text{[AP]} + 1\text{[AP]} = 3\text{[AP]}$.

Now the OP was wondering about $2 + 2/2$, and it might be useful for them to review mixed number notation. But

$\tag 4 \frac{2}{2} = 2 \times \frac{1}{2} = \frac{1}{2} + \frac{1}{2} = 1$

(right - combining half a pie with half a pie is a full pie)

So $2 + 2/2 = 3$.

Also, $2 \times 2/2 = \frac{2}{2} + \frac{2}{2} = (\frac{1}{2} + \frac{1}{2}) + (\frac{1}{2} + \frac{1}{2}) = 1 + 1 = 2$

After the OP is totally comfortable with these ideas and can use them without a second thought, they can proceed to understanding multiplication and division.