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From @fresh_42's Insight

https://www.physicsforums.com/insights/10-math-things-we-all-learnt-wrong-at-school/

Please discuss!

They are not. They are equivalence classes. My favorite example is, that it makes a huge difference whether you carry home a pie from the bakery or ##12/12## pieces of a pie. The amount of pie and the prizes would be the same, their appearance is not. Of course, we treat ##1=\frac{12}{12}## the same because we are interested in its value, however, they are only equal because ##1\cdot 12 = 12 \cdot 1.## It becomes clearer in its general form:

$$\dfrac{a}{b}\sim\dfrac{c}{d}\Longleftrightarrow a\cdot d= b\cdot c$$

'##\sim##' is strictly speaking an equivalence relation. It gathers really many quotients under one name

$$1=\left\{1,\dfrac{2}{2},\dfrac{-3}{-3},\dfrac{12}{12},\ldots\right\}$$

and the same is true for all other quotients. We take them as the same and write '##=##' instead of '##\sim##' because we are only interested in their values. But ##1\neq \dfrac{12}{12}.## You can literally see that it is different: ##5## symbols instead of ##1.##

https://www.physicsforums.com/insights/10-math-things-we-all-learnt-wrong-at-school/

Please discuss!

They are not. They are equivalence classes. My favorite example is, that it makes a huge difference whether you carry home a pie from the bakery or ##12/12## pieces of a pie. The amount of pie and the prizes would be the same, their appearance is not. Of course, we treat ##1=\frac{12}{12}## the same because we are interested in its value, however, they are only equal because ##1\cdot 12 = 12 \cdot 1.## It becomes clearer in its general form:

$$\dfrac{a}{b}\sim\dfrac{c}{d}\Longleftrightarrow a\cdot d= b\cdot c$$

'##\sim##' is strictly speaking an equivalence relation. It gathers really many quotients under one name

$$1=\left\{1,\dfrac{2}{2},\dfrac{-3}{-3},\dfrac{12}{12},\ldots\right\}$$

and the same is true for all other quotients. We take them as the same and write '##=##' instead of '##\sim##' because we are only interested in their values. But ##1\neq \dfrac{12}{12}.## You can literally see that it is different: ##5## symbols instead of ##1.##

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