How To Really Understand Units In Mathematics - An illustration showing two neatly stacked piles of 6 apples, one on each side of the illustration. At the centre, you see the following question: "Which is correct?" Below this, there are two sentences: 1. A dozen apples is rotten. 2. A dozen apples are rotten.

Before we understand units in mathematics, let us start with units in real life. At this point, you might be thinking:

“Hang on a second. Where do we use units in real life?”

I hear you. The answer to that question is that we use units in real life all the time. For example, think about the last time you weighed yourself on your weighing scale. Let us say that the number came out in Kilograms.

A ‘kilogram’ is a unit. But then, other people in other parts of the world measure their weights in pounds. So, what gives? Why do we have two separate units for the same thing?

There’s something fishy going on here, right? Oh, but wait! It gets worse. Let us now suppose that someone convinced everyone to measure their weights in kilograms.

At this point, a chef would object to it whilst wagging his finger. All his life, he has measured the weight of his food ingredients in grams, not kilograms! Are you mildly frustrated at this point? Let me push that button one more time.

Imagine that you bought 12 apples from your local grocery store. To your horror, by the time you reach home, you find out that all 12 apples are rotten. Given this situation, which of the following sentences is correct?

1. A dozen apples is rotten.

2. A dozen apples are rotten.

I will get to the answer to this question by the end of this essay. But for now, this question leads us to the fundamentals of units.

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The Fundamentals of Units in Real Life

You see, the term ‘dozen’ might not sound like ‘gram’ or ‘kilogram’ but it is very much a unit (in the mathematical sense) as well.

Let us assume that your apples are not rotten. You take one apple out of your bag containing a dozen apples and cut it into a dozen slices.

Then, you proceed to eat one apple slice. Here, one slice, one apple, and a (one) dozen are all units. They just refer to different things. That is a fundamental property of units.

They help us refer to the quantity of stuff in different contexts. You can create units using two approaches:

1. Partitioning

2. Combining

How to Partition and Combine to Create New Units

I’ll save you the boredom of literal explanations and proceed with examples. ‘A dozen’ is an example of a combined unit. We need twelve apples to make a dozen.

A slice is an example of a partitioned unit. An apple can be sliced (partitioned) into ’n’ slices.

The sweet thing about combined and partitioned units is that they can be combined and/or partitioned again and again to make up new units.

How To Really Understand Units In Mathematics — An illustration showing two neatly stacked piles of 6 green apples. In total, there are 12 apples. In other words, there are a dozen apples.
A dozen apples — Illustrative art created by the author

For instance, you could cut a slice into twelve pieces yet again. Or you could collect 100 apple slices, put then into a box, and call it a box of apple slices.

All of this sounds very elementary, isn’t it? But I am going through this discussion for a reason. You see, these concepts link directly how units work in mathematics as well.

How to Really Understand Units in Mathematics?

Inmathematics, we don’t have different units like kilograms or pounds, right? Well, not really. Consider the number ‘1’. We all know that 1 equals 1.

However, we could combine (add) a bunch of 1s to make another number:

1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 = 10

The resulting number is what we refer to as ‘ten’. But note that this ‘ten’ has a ‘1’ in it as well. What is this ‘1’ doing there? You might remember from childhood that this ‘1’ occupies the ten’s place (in the decimal system).

What this actually means is that the ‘1’ in ‘10’ is a unit that quantifies the number of tens in this number. It is no different from a dozen or a kilogram, functionally speaking.

We could partition the 10 into ten 1s or combine ten 10s to make 100. Note that the ‘1’ in ‘100’ is a unit quantifying the number of hundreds in this number.

We can clearly see that the logic of partitioning and combining to create new units works just as well in mathematics as it does in the real world. The only difference is that this happens at a lower level of abstraction.

In the real world, we could choose to collect 100 apple slices in a box, and refer to it as ‘1’ box of apple slices; the same ‘1’ as before but referring to something completely different in a completely different context.

Final Comments

We now come back to the question from the introduction of this essay. Which of the following sentences is correct?

1. A dozen apples is rotten.

2. A dozen apples are rotten.

How To Really Understand Units In Mathematics — An illustration showing two neatly stacked piles of 6 apples, one on each side of the illustration. At the centre, you see the following question: “Which is correct?” Below this, there are two sentences: 1. A dozen apples is rotten. 2. A dozen apples are rotten.
Which is correct? — Illustration created by the author

The answer: “A dozen apples are rotten”. The reason is actually not purely mathematical; English grammar is involved as well. A “dozen” is a collective noun, just like a “box” of apple slices.

In the case of the box, you would say “The box of apple slices is broken.” because we refer to the box directly. However, ‘dozen’ works like a number in this context, in that, we refer to ‘a dozen apples’ just like ’12 apples’.

Having said that a dozen can also be singular based on a collective context. Don’t shoot the messenger here. English grammar can be complicated like that sometimes.

Now I bet that you are glad that this essay was about mathematics and not English grammar. We like to keep things simple around this corner of the internet.

Reference and credit: Christopher Danielson.

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