# Zero/Zero questions and perhaps faulty logic So I only have an Algebra II level understanding of math seeing as I am still in high school and am still missing some fundamentals seeing as I didn't pay attention in math until this year. However when recalling something my algebra teacher had taught me during the year I came up with some questions regarding the logic recently. So during the school year, I was taught that 2/2=1,a/a=1,(xy)/(xy)=1 and so forth but 0/0=Undefined... and while researching this topic I found that the algebraic way to write all these fractions is as such 2(x)=2,a(x)=a, and 0(x)=0 and upon researching this further I found that the reason that 0/0 is undefined is that for any value of x the equation holds true. However, seeing as in the fraction a/a a is a variable and

Zero/Zero questions and perhaps faulty logic
So I only have an Algebra II level understanding of math seeing as I am still in high school and am still missing some fundamentals seeing as I didn't pay attention in math until this year. However when recalling something my algebra teacher had taught me during the year I came up with some questions regarding the logic recently.
So during the school year, I was taught that $\frac{2}{2}=1,\frac{a}{a}=1,\frac{xy}{xy}=1$ and so forth but $\frac{0}{0}=\text{Undefined}$... and while researching this topic I found that the algebraic way to write all these fractions is as such $2\left(x\right)=2,a\left(x\right)=a,$ and upon researching this further I found that the reason that $\frac{0}{0}$ is undefined is that for any value of $x$ the equation holds true. However, seeing as in the fraction $\frac{a}{a}$ $a$ is a variable and variables can represent any given quantity I was wondering in the case that $a=0$ would $\frac{a}{a}$ still $=1$ and if not why along with the fact that lets say $a=0$ and you didn't know it why is it safe to assume that $a$ would never equal zero? Also if it happens to be the case where when $a=0,\frac{a}{a}=1$ (which I doubt it is) shouldn't this mean that $\frac{0}{0}=1$ then?
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Raul Garrett
$\frac{a}{a}=\left\{\begin{array}{ll}1& a\ne 0\\ \text{undefined}& a=0\end{array}$
Strictly speaking, you have to check whether $a$ is zero before you cancel terms off from both numerator or denominator.
Even before we write down $\frac{a}{a}$, we should first check if the denominator can be zero, i.e. check it before we write down such term.
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