A thread on the mathematical concepts of fractions and wholeness.

3 thoughts on “Fractions and Wholeness”

(I wrote some notes on this topic a few months ago.)

Fractions and Wholeness
To understand the concept of fractions in math, one has to understand the concept of wholeness, which, in math, is represented by whole numbers. My sense is that the root or primary whole number is the number 1. It represents wholeness.* The number 1 is totally complete. Any whole number above the number 1 is basically a multiple(?) of 1. For example, the number 5 can be thought of as five wholes or five 1s. 5 is five complete, whole quantities. The whole numbers above 1 are many, “plural.” “1” is the only number, as far as I know, that is single, but not zero or a fraction. (-1 is essentially the same as +1. ?)
Fractions are a fragment of a whole or “1” and therefore incomplete. They take a whole or a group of wholes and break them into equal parts (usually) and then select one or more in relation to all the parts. For example, an example of a 1/3 would be taking a chocolate bar, breaking it into three pieces and selecting one piece. The one piece, in relation to whole chocolate bar, is 1/3. Fractions are less than 1 but greater than 0. They are a quantity that exists between nothingness and wholeness/completeness/unity (1).
Whole numbers, by virtue of being whole, are complete, compared to fractions. Fractions seem to only exist in relation to a whole or 1. Whole numbers can exist without fractions, but fractions cannot exist without whole numbers—specifically the number 1. (Is this accurate? Can we really understand “wholeness” without understanding the concept of fractions?)
Fractions seem more like a relationship more than an actual quantity. (Actually, that seems wrong. If I have 5.5 apple pies, half a pie is an actual quantity. At the same time, suppose I had 5.5 apple pies that were 6” in diameter and 5.5 pies that were 1’ in diameter. The half pies from both groups are not equal. At the same time the whole pies are not equal, too. In this case, 5.5 only refers to the number of pies, not the weight or volume. The numbers for the latter would be different.) A way to see this more clearly is to think of a percentages, which are essentially fractions (based on a denominator of 100). Getting a payment based on a specific dollar amount (e.g., $1,000) versus a percentage of profits is not the same at all. Percentages or fractions seem less set or specific. Whole numbers have a specificity and exactness that fractions do not; fractions are relative in a way that whole numbers are not. (Not sure if this is right.)
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*One should be careful about thinking of the number 1 and wholeness synonymously. “1” can represent wholeness, but wholeness is not always equivalent to the number 1. For example, let’s say you cut 1 chocolate bar into three equal pieces. One of those pieces is 1/3 of the bar. But that piece can also be thought of or “become” a whole as well. It can both be 1/3 of the bar or 1 (whole unto itself). To see this, suppose I took the 1/3 piece and decided to share it with a friend, breaking the piece in two equal halves. We could think of the piece as a whole being divided into two. Or we could think of it as 1/3 being divided into two. Wholeness or fractiousness depends on perspective. Something can be viewed as a whole or fraction depending on what we want to do.

The concept of 1, multiples of 1, and parts of 1 is pretty key. If this works for you I think you should go with it. You’ll see it a lot in trig too, which you referenced when talking about pi last weekend.

If you can get to fractions of 1 being the same thing as multiples of 1, this will all be a piece of cake for you.

You’ll see it a lot in trig too, which you referenced when talking about pi last weekend.

I have a question about pi: Should we consider it as a fraction? To me, it seems more like a ratio. I suspect both ratios are fractions, but I’m distinguishing fractions as the relationship between a part and whole. That is, you cut up or divide something into a certain number of pieces and then you select one or more of those pieces.

A ratio, on the other hand, doesn’t involve dividing or cutting up something–or is that wrong? Instead, it involves a relationship between one aspect of something to another aspect of that thing. For pi, the relationship involves the circumference and the diameter, not really a relationship between parts of a whole. These seems very different from cutting up a pizza or dividing a pot of money. Circumference/diameter is a fraction, but it seems significantly different from other pieces of a pizza relating to the whole.

On a related note, can we/should we think of pi as 314.(non-terminating decimals)%? That is, the circumference of a circle is always 314% of the diameter?

On an unrelated note–and I think this relates to a point you made about integers–I’m wondering if we should think of numbers (including fractions) in a similar way we should view words–namely, that they’re representations not the thing in and of itself. For example, “giraffe” is a word we use when we refer to a long-necked, spotted, African equine animal. That may seem obvious, but I think people can lose sight of this point. I think I may have been losing sight of the fact that numbers are not really a concrete thing; they’re far more abstract than I might have realized. And I think this explains, at least in part, the trouble I had when thinking about what fractions really are.

Another problem may have occurred because I failed to appreciate the limitations of numbers (and other mathematical concepts). What I’m suggesting is that math, like language, has limitations; that is, they are imperfect and can’t fully capture whatever it is they’re used for. In some or many situations, the numbers may fully capture whatever is being measured or described. But in other ways or situations, numbers may have “holes” so to speak. I may be thinking of numbers as if they don’t have these holes.

(I wrote some notes on this topic a few months ago.)

Fractions and Wholeness

To understand the concept of fractions in math, one has to understand the concept of wholeness, which, in math, is represented by whole numbers. My sense is that the root or primary whole number is the number 1. It represents wholeness.* The number 1 is totally complete. Any whole number above the number 1 is basically a multiple(?) of 1. For example, the number 5 can be thought of as five wholes or five 1s. 5 is five complete, whole quantities. The whole numbers above 1 are many, “plural.” “1” is the only number, as far as I know, that is single, but not zero or a fraction. (-1 is essentially the same as +1. ?)

Fractions are a fragment of a whole or “1” and therefore incomplete. They take a whole or a group of wholes and break them into equal parts (usually) and then select one or more in relation to all the parts. For example, an example of a 1/3 would be taking a chocolate bar, breaking it into three pieces and selecting one piece. The one piece, in relation to whole chocolate bar, is 1/3. Fractions are less than 1 but greater than 0. They are a quantity that exists between nothingness and wholeness/completeness/unity (1).

Whole numbers, by virtue of being whole, are complete, compared to fractions. Fractions seem to only exist in relation to a whole or 1. Whole numbers can exist without fractions, but fractions cannot exist without whole numbers—specifically the number 1. (Is this accurate? Can we really understand “wholeness” without understanding the concept of fractions?)

Fractions seem more like a relationship more than an actual quantity. (Actually, that seems wrong. If I have 5.5 apple pies, half a pie is an actual quantity. At the same time, suppose I had 5.5 apple pies that were 6” in diameter and 5.5 pies that were 1’ in diameter. The half pies from both groups are not equal. At the same time the whole pies are not equal, too. In this case, 5.5 only refers to the number of pies, not the weight or volume. The numbers for the latter would be different.) A way to see this more clearly is to think of a percentages, which are essentially fractions (based on a denominator of 100). Getting a payment based on a specific dollar amount (e.g., $1,000) versus a percentage of profits is not the same at all. Percentages or fractions seem less set or specific. Whole numbers have a specificity and exactness that fractions do not; fractions are relative in a way that whole numbers are not. (Not sure if this is right.)

—

*One should be careful about thinking of the number 1 and wholeness synonymously. “1” can represent wholeness, but wholeness is not always equivalent to the number 1. For example, let’s say you cut 1 chocolate bar into three equal pieces. One of those pieces is 1/3 of the bar. But that piece can also be thought of or “become” a whole as well. It can both be 1/3 of the bar or 1 (whole unto itself). To see this, suppose I took the 1/3 piece and decided to share it with a friend, breaking the piece in two equal halves. We could think of the piece as a whole being divided into two. Or we could think of it as 1/3 being divided into two. Wholeness or fractiousness depends on perspective. Something can be viewed as a whole or fraction depending on what we want to do.

The concept of 1, multiples of 1, and parts of 1 is pretty key. If this works for you I think you should go with it. You’ll see it a lot in trig too, which you referenced when talking about pi last weekend.

If you can get to fractions of 1 being the same thing as multiples of 1, this will all be a piece of cake for you.

I have a question about pi: Should we consider it as a fraction? To me, it seems more like a ratio. I suspect both ratios are fractions, but I’m distinguishing fractions as the relationship between a part and whole. That is, you cut up or divide something into a certain number of pieces and then you select one or more of those pieces.

A ratio, on the other hand, doesn’t involve dividing or cutting up something–or is that wrong? Instead, it involves a relationship between one aspect of something to another aspect of that thing. For pi, the relationship involves the circumference and the diameter, not really a relationship between parts of a whole. These seems very different from cutting up a pizza or dividing a pot of money. Circumference/diameter is a fraction, but it seems significantly different from other pieces of a pizza relating to the whole.

On a related note, can we/should we think of pi as 314.(non-terminating decimals)%? That is, the circumference of a circle is always 314% of the diameter?

On an unrelated note–and I think this relates to a point you made about integers–I’m wondering if we should think of numbers (including fractions) in a similar way we should view words–namely, that they’re representations not the thing in and of itself. For example, “giraffe” is a word we use when we refer to a long-necked, spotted, African equine animal. That may seem obvious, but I think people can lose sight of this point. I think I may have been losing sight of the fact that numbers are not really a concrete thing; they’re far more abstract than I might have realized. And I think this explains, at least in part, the trouble I had when thinking about what fractions really are.

Another problem may have occurred because I failed to appreciate the limitations of numbers (and other mathematical concepts). What I’m suggesting is that math, like language, has limitations; that is, they are imperfect and can’t fully capture whatever it is they’re used for. In some or many situations, the numbers may fully capture whatever is being measured or described. But in other ways or situations, numbers may have “holes” so to speak. I may be thinking of numbers as if they don’t have these holes.