Several good answers already, but there is another way to look at it.

The actual, formal definition of a concept like this is going to be based on axioms. Those basically just define equality ($\frac{a}{b} = \frac{c}{d}$ if $ad = bc$), addition, subtraction, multiplication and division in terms of the natural numbers. Then we prove that we haven't made arithmetic contradict itself just now, if it doesn't already, by extending it from whole numbers to fractions. We also want to prove that our operations have the properties that most theorems rely on, such as the order in which we add numbers not mattering.

But we teach kids about fractions (or rational numbers) and consider the octonions an interesting curiosity that you'll learn about if you get a degree in math because the fractions are useful in daily life. And complex numbers and quaternions are in between because we need them to solve certain problems in engineering.

So, on one level, the answer might be something like: $3 \div 4$ is $\frac{3}{1} \times \frac{1}{4} = \frac{3}{4}$. But you knew that already. If some set of things corresponds to fractions, that relationship will hold true. Otherwise, you'd have shown that the things that contradict our theorems about fractions are not like fractions. If you're really asking why that corresponds to a couple different things in the real world, a geometric proof like Henry's is a great approach.

Several good answers already, but there is another way to look at it.

The actual, formal definition of a concept like this is going to be based on axioms. Those basically just define equality ($\frac{a}{b} = \frac{c}{d}$ if $ad = bc$), addition, subtraction, multiplication and division in terms of the natural numbers. Then we prove that we haven't made arithmetic contradict itself just now, if it hadn't already, by extending it from whole numbers to fractions. We also want to prove that our operations have the properties that most theorems rely on, such as the order in which we add numbers not mattering.

But we teach kids about fractions (or rational numbers) and consider the octonions an interesting curiosity that you'll learn about if you get a degree in math because the fractions are useful in daily life. And complex numbers and quaternions are in between because we need them to solve certain problems in engineering.

So, on one level, the answer might be something like: $3 \div 4$ is $\frac{3}{1} \times \frac{1}{4} = \frac{3}{4}$. But you knew that already. If some set of things corresponds to fractions, that relationship will hold true. Otherwise, you'd have shown that the things that contradict our theorems about fractions are not like fractions. If you're really asking why that corresponds to a couple different things in the real world, a geometric proof like Henry's is a great approach.

Several good answers already, but there is another way to look at it.

The actual, formal definition of a concept like this is going to be based on axioms. Those basically just define equality ($\frac{a}{b} = \frac{c}{d}$ if $ad = bc$), addition, subtraction, multiplication and division in terms of the natural numbers. Then we prove that we haven't made arithmetic contradict itself just now, if it doesn't already, by extending it from whole numbers to fractions. We also want to prove that our operations have the properties that most theorems rely on, such as the order in which we add numbers not mattering.

But we care about fractions (or rational numbers) and consider the octonions an interesting curiosity because the fractions are useful in daily life. And complex numbers and quaternions are in between because we need them to solve certain problems in engineering.

So, on one level, the answer might be something like: $3 \div 4$ is $\frac{3}{1} \times \frac{1}{4} = \frac{3}{4}$. But you knew that already. If some set of things corresponds to fractions, that relationship will hold true. Otherwise, you'd have shown that the things that contradict our theorems about fractions are not like fractions. If you're really asking why that corresponds to a couple different things in the real world, a geometric proof like Henry's is a great approach.

Several good answers already, but there is another way to look at it.

The actual, formal definition of a concept like this is going to be based on axioms. Those basically just define equality ($\frac{a}{b} = \frac{c}{d}$ if $ad = bc$), addition, subtraction, multiplication and division in terms of the natural numbers. Then we prove that we haven't made arithmetic contradict itself just now, if it doesn't already, by extending it from whole numbers to fractions. We also want to prove that our operations have the properties that most theorems rely on, such as the order in which we add numbers not mattering.

But we teach kids about fractions (or rational numbers) and consider the octonions an interesting curiosity that you'll learn about if you get a degree in math because the fractions are useful in daily life. And complex numbers and quaternions are in between because we need them to solve certain problems in engineering.

So, on one level, the answer might be something like: $3 \div 4$ is $\frac{3}{1} \times \frac{1}{4} = \frac{3}{4}$. But you knew that already. If some set of things corresponds to fractions, that relationship will hold true. Otherwise, you'd have shown that the things that contradict our theorems about fractions are not like fractions. If you're really asking why that corresponds to a couple different things in the real world, a geometric proof like Henry's is a great approach.