So, really basic mathematics question, but would the fraction $2/2$ be the exact same as dividing $2$ by $2$? Would this apply to all fractions and all division problems? Or is it just coincidence? I'm not entirely sure and I get vague answers from my mathematics teacher, so I'm not $100$%.
I figured I'd ask this community to see what everyone thought.
Yes, when we write a fraction like $$\frac{a}{b}$$ this means exactly the same thing as "the result of dividing $a$ by $b$". For example, $2 \div 3$ is precisely equal to $\frac{2}{3}$, $1 \div 5$ is equal to $\frac{1}{5}$, and $10 \div 2$ is equal to $\frac{10}{2}$, which can also be written as $\frac{5}{1}$ or simply as $5$.
This idea -- that division and fractions are essentially the same idea -- is one that many students seem so struggle with, perhaps precisely because it's so fundamental. But it comes in extremely handy when dealing with more complicated expressions, such as fractions that have other fractions nested within them. For example, suppose you are confronted with an expression like $$\frac{\frac{24}{7}}{\frac{12}{35}}$$ If you remember that the "main" fraction bar just means division, then this is the same as $$\frac{24}{7} \div \frac{12}{35}$$ But this, in turn, is the same thing as $$\frac{24}{7} \times \frac{35}{12}$$ (If you are not sure about dividing one fraction by another, see https://matheducators.stackexchange.com/a/7868/29)
which in turn can be simplified down to just $$\require{cancel} \frac{ 2 \times \cancel{12} \times 5 \times \cancel{7}}{\cancel{7} \times \cancel{12}}=10$$