Return to Answer

The thing missing from the units given in examples (m, C, etc) in multiple dimensions is a qualifier of which dimension. For example, when we identify an area and give it a unit, say $m^2$, we really mean $m$ in direction x times $m$ in direction perpendicular to x. When we say $C^2$ (Coulomb), we mean $C$ of one entity times $C$ of a different entity. These important qualifiers ("of one entity" and "of a different entity") disappear - become implicit - when we multiply the quantities and assign units.

We can, of course, have $C^2$ refer to the same apparent entity, like the nucleus of an atom (e.g. $Z^2$), but, really, terms like these still imply separate charges existing within the nucleus, or sometimes the charge of an electron interacting with the charge of a nucleus, or sometimes just are indirect but convenient references by proportionality to the surface area, volume, or various symmetries of nuclear constituents (https://en.wikipedia.org/wiki/Semi-empirical_mass_formula).

So, for apples, only if there is a quantity proportional to apple would a unit be derived. For example, the quantity of applesauce created when smashing apples together inside a piston is proportional to the volume of apples, which is proportional to the number of apples. So $Q \propto A$. But the rate at which applesauce is created by smashing apples together in a shaking machine (?) is proportional to the rate of collision of (different) apples, which is $R \propto A^2$. (See reaction rate of colliding particles in a population of particles, so called "Collision Theory") Here, each $A$ in $A^2$ is implicitly referring to one of each apple in a pair (different apples).

The reason we don't keep these qualifiers of units or, in the case of numbering objects, units themselves, is mere convenience. A perfectly precise description would include the unit apple, and for that matter which apple. Just as we accept that frequency has a unit $s^{-1}$ and leave it to the reader to understand that we really mean $\frac{\text{cycles of whatever wave we're discussing}}{s}$.

The thing missing from the units given in examples (m, C, etc) in multiple dimensions is a qualifier of which dimension. For example, when we identify an area and give it a unit, say $m^2$, we really mean $m$ in direction x times $m$ in direction perpendicular to x. When we say $C^2$ (Coulomb), we mean $C$ of one entity times $C$ of a different entity. These important qualifiers ("of one entity" and "of a different entity") disappear - become implicit - when we multiply the quantities and assign units.

We can, of course, have $C^2$ refer to the same apparent entity, like the nucleus of an atom (e.g. $Z^2$), but, really, terms like these still imply separate charges existing within the nucleus, or sometimes the charge of an electron interacting with the charge of a nucleus, or sometimes just are indirect but convenient references by proportionality to the surface area, volume, or various symmetries of nuclear constituents (https://en.wikipedia.org/wiki/Semi-empirical_mass_formula).

So, for apples, only if there is a quantity proportional to apple would a unit be derived. For example, the quantity of applesauce created when smashing apples together inside a piston is proportional to the volume of apples, which is proportional to the number of apples. So $Q \propto A$. But the rate at which applesauce is created by smashing apples together in a shaking machine (?) is proportional to the rate of collision of (different) apples, which is $R \propto A^2$. (See reaction rate of colliding particles in a population of particles, so called "Collision Theory") Here, each $A$ in $A^2$ is implicitly referring to one of each apple in a pair (different apples).

The reason we don't keep these qualifiers of units or, in the case of numbering objects, units themselves, is mere convenience; however, when performing dimensional analysis, it may be extremely important to keep the qualifiers so as not to mistake a green apple for a red. A perfectly precise description would include the unit apple, and for that matter which apple. Just as we accept that frequency has a unit $s^{-1}$ and leave it to the reader to understand that we really mean $\frac{\text{cycles of whatever wave we're discussing}}{s}$.

The thing missing from the units given in examples (m, C, etc) in multiple dimensions is a qualifier of which dimension. For example, when we identify an area and give it a unit, say $m^2$, we really mean $m$ in direction x times $m$ in direction perpendicular to x. When we say $C^2$ (Coulomb), we mean $C$ of one entity times $C$ of a different entity. These important qualifiers ("of one entity" and "of a different entity") disappear - become implicit - when we multiply the quantities and assign units.

We can, of course, have $C^2$ refer to the same apparent entity, like the nucleus of an atom (e.g. $Z^2$), but, really, terms like these still imply separate charges existing within the nucleus, or sometimes the charge of an electron interacting with the charge of a nucleus, or sometimes just are indirect but convenient references by proportionality to the surface area, volume, or various symmetries of nuclear constituents (https://en.wikipedia.org/wiki/Semi-empirical_mass_formula).

So, for apples, only if there is a quantity proportional to apple would a unit be derived. For example, the quantity of applesauce created when smashing apples together inside a piston is proportional to the volume of apples, which is proportional to the number of apples. So $Q \propto A$. But the rate at which applesauce is created by smashing apples together in a shaking machine (?) is proportional to the rate of collision of (different) apples, which is $R \propto A^2$. (See reaction rate of colliding particles in a population of particles, so called "Collision Theory") Here, each $A$ in $A^2$ is implicitly referring to one of each apple in a pair (different apples).

The reason we don't keep these qualifiers of units or, in the case of numbering objects, units themselves, is mere convenience. A perfectly precise description would include the unit apple, and for that matter which apple.

The thing missing from the units given in examples (m, C, etc) in multiple dimensions is a qualifier of which dimension. For example, when we identify an area and give it a unit, say $m^2$, we really mean $m$ in direction x times $m$ in direction perpendicular to x. When we say $C^2$ (Coulomb), we mean $C$ of one entity times $C$ of a different entity. These important qualifiers ("of one entity" and "of a different entity") disappear - become implicit - when we multiply the quantities and assign units.

We can, of course, have $C^2$ refer to the same apparent entity, like the nucleus of an atom (e.g. $Z^2$), but, really, terms like these still imply separate charges existing within the nucleus, or sometimes the charge of an electron interacting with the charge of a nucleus, or sometimes just are indirect but convenient references by proportionality to the surface area, volume, or various symmetries of nuclear constituents (https://en.wikipedia.org/wiki/Semi-empirical_mass_formula).

So, for apples, only if there is a quantity proportional to apple would a unit be derived. For example, the quantity of applesauce created when smashing apples together inside a piston is proportional to the volume of apples, which is proportional to the number of apples. So $Q \propto A$. But the rate at which applesauce is created by smashing apples together in a shaking machine (?) is proportional to the rate of collision of (different) apples, which is $R \propto A^2$. (See reaction rate of colliding particles in a population of particles, so called "Collision Theory") Here, each $A$ in $A^2$ is implicitly referring to one of each apple in a pair (different apples).

The reason we don't keep these qualifiers of units or, in the case of numbering objects, units themselves, is mere convenience. A perfectly precise description would include the unit apple, and for that matter which apple. Just as we accept that frequency has a unit $s^{-1}$ and leave it to the reader to understand that we really mean $\frac{\text{cycles of whatever wave we're discussing}}{s}$.

The thing missing from the units given in examples (m, C, etc) in multiple dimensions is a qualifier of which dimension. For example, when we identify an area and give it a unit, say $m^2$, we really mean $m$ in direction x times $m$ in direction perpendicular to x. When we say $C^2$ (Coulomb), we mean $C$ of one entity times $C$ of a different entity. These important qualifiers ("of one entity" and "of a different entity") disappear - become implicit - when we multiply the quantities and assign units.

We can, of course, have $C^2$ refer to the same apparent entity, like the nucleus of an atom (e.g. $Z^2$), but, really, terms like these still imply separate charges existing within the nucleus, or sometimes the charge of an electron interacting with the charge of a nucleus, or sometimes just are indirect but convenient references by proportionality to the surface area, volume, or various symmetries of nuclear constituents (https://en.wikipedia.org/wiki/Semi-empirical_mass_formula).

So, for apples, only if there is a quantity proportional to apple would a unit be derived. For example, the quantity of applesauce created when smashing apples together inside a piston is proportional to the volume of apples, which is proportional to the number of apples. So $Q \propto A$. But the rate at which applesauce is created by smashing apples together in a shaking machine (?) is proportional to the rate of collision of (different) apples, which is $R \propto A^2$. (See reaction rate of colliding particles in a population of particles, so called "Collision Theory") Here, each $A$ in $A^2$ is implicitly referring to one of each apple in a pair (different apples).

The thing missing from the units given in examples (m, C, etc) in multiple dimensions is a qualifier of which dimension. For example, when we identify an area and give it a unit, say $m^2$, we really mean $m$ in direction x times $m$ in direction perpendicular to x. When we say $C^2$ (Coulomb), we mean $C$ of one entity times $C$ of a different entity. These important qualifiers ("of one entity" and "of a different entity") disappear - become implicit - when we multiply the quantities and assign units.

We can, of course, have $C^2$ refer to the same apparent entity, like the nucleus of an atom (e.g. $Z^2$), but, really, terms like these still imply separate charges existing within the nucleus, or sometimes the charge of an electron interacting with the charge of a nucleus, or sometimes just are indirect but convenient references by proportionality to the surface area, volume, or various symmetries of nuclear constituents (https://en.wikipedia.org/wiki/Semi-empirical_mass_formula).

So, for apples, only if there is a quantity proportional to apple would a unit be derived. For example, the quantity of applesauce created when smashing apples together inside a piston is proportional to the volume of apples, which is proportional to the number of apples. So $Q \propto A$. But the rate at which applesauce is created by smashing apples together in a shaking machine (?) is proportional to the rate of collision of (different) apples, which is $R \propto A^2$. (See reaction rate of colliding particles in a population of particles, so called "Collision Theory") Here, each $A$ in $A^2$ is implicitly referring to one of each apple in a pair (different apples).

The reason we don't keep these qualifiers of units or, in the case of numbering objects, units themselves, is mere convenience. A perfectly precise description would include the unit apple, and for that matter which apple.