The Magic of Numbers – Serial 7

Recalling the sizes of things

As you grow up, you see more things, that means you recognize more sizes and are able to identify them. As you see each new object, you compare them to the size of your body. You use comparative vocabulary to differentiate the sizes of things, similar to differentiating spacings of trees, as we did it in the first serial. Only then you can remember the sizes.

You see objects such as people, basketballs, televisions, boats, computers, waterfalls and skyscrapers. You know which one is big or small by comparing them to the size of your body. Not only that, you can also compare all those objects listed with the size of a basketball. We don’t care how those objects were made, we just see and grew up as children with those objects.

Rating Systems: The precursor to measuring devices

You learnt how to count using the Hindu-Arabic number writing system. As you were counting with the sequence of numbers you were taught with, you feel that as you count more of the same objects, things feel bigger based on your immediate vision field.

You notice that the class blackboard only has limited amount of space to show pictures. Based on the sequence of numbers, you were taught the questions “How many?” or “How much?”. Counting, which is also called quantifying or enumerating is based on these questions. The mechanism (pattern) of counting is discussed in Serial 2.

Your teacher or parents has shown you how big objects are. They teach you how to use a rating system based on the sizes of objects they have shown to you. This rating system is based on English numerals and Hindu-Arabic number symbols based on the proper sequence of those numbers.

The teacher uses a rating system that has five stars, based on the sizes of the objects and the proper sequence of the numerals, she assigns numerals to the objects: “You can rate the sizes of any new object based on these reference points. Comparative vocabulary is useful for it”.

Absolute (big or small), comparative (bigger or smaller), and superlative (biggest or smallest) were already taught as adjectives in the English class. But often it feels like English class is not related to Maths class.

We see that a basketball is small, we rate it as one (1) star. We can call this small.

We see that the computer is bigger than a basketball, but smaller than a baby’s crib, we rate it as two (2) stars. We can call this quite small.

We see that a baby’s crib is bigger than a computer, but smaller than an oak tree, so we rate it as three (3) stars. We can call this as medium size.

We see that an oak tree is bigger than a baby’s crib, but smaller than your house, so we rate it as four (4) stars. We can call this quite big.

We see that house is big, we rate is as five (5) stars. We can call this big.

So you say to yourself some superlatives: The basketball is the smallest object I have ever seen and the house is the biggest object I have ever seen. The rating system is also illustrated below:

So the everyday vocabulary such as small, quite small, medium, quite big and big can be augmented with a star rating system. So when we rate any new object, let’s say a television, we say it is quite small.

Saying it as quite big means we are comparing to the sizes in the list. This also means it is roughly the same size as a computer, but different from the other objects in the list. Different can mean bigger or smaller.

As you grow up you get to know and recognize more objects. You saw a book, and recalling the sizes of the basketball and a computer, you conclude the size of the book is just in between a basketball and a computer. You have trouble rating it.

Or you saw an object smaller than a basketball, it can be a toy marble or a computer mouse. You also saw directly in your eyes that a toy marble is smaller than a computer mouse. You don’t even need to think of the size because you simply saw them within your vision field. You conclude those things were very small and the toy marble is the smallest thing you have ever seen.

Or you went in front of a skyscraper or a lake, as you saw them, you recall from your memory the size of your house. You remember that they are both bigger than your house. Seeking patterns, you comment that they are the biggest things you have ever seen. You knew they are beyond the rating system you have set up.

So what do you do? You find that it is difficult to remember the sizes of all things. A solution to this is in the next section. To make it easier to remember, you select a common comparison point for all objects, say the size of basketball. This is similar to you selecting the spacing between the apple tree and the cherry tree in the first section, enough to make sense (of scale) of the others.

Many books about skyscrapers or dinosaurs have diagrams and infographics intended to give you a sense of scale: They put the buildings side by side and might even include a person. This strategy helps you compare the skyscrapers and also compare them with the size of your body.

Units

Since there are many objects and distances (lengths) to remember, it is very convenient for the human mind to simply think of one reference point to make sense of all the others. This was described in the last section of Serial 1. We can call the reference point we chose as a unit. Units helps us when numbers and counting does the job better than using comparative vocabulary.

So in the previous section, we simply assigned numbers to objects big and small. We can also assign the sequence of the Latin alphabet but the characters (26) are limited, the continuum of numbers is infinite. Smaller objects (1 star) are closer to the beginning of the number sequence and larger objects are much further from it (5 star).

An authority has decided to use the size of the foot as common point of comparison for all things. They engrave the length from the biggest toenail to the sole of the foot on a stone tablet, but they make it slightly longer than the length of the foot. They call it as 1 feet.

They announced this unit via the mass media on your television and you were taught how to use it in school. Below is graphic of the front of the stone block. Next two sections will demonstrate how measuring devices are made.

The feet is called as a unit. Units are defined as a common point of comparison, that means you can compare everything with it. We can use this unit to determine how big or long that object is. This convenient if you don’t like carrying basketballs all day just to compare the sizes of things with it.

Measurement is similar to counting ordinary objects.

We use our ability to detect patterns (color) in the objects we see. We differentiate between those patterns by associating a word (sound from our voice box). Differentiating between feet and meters is just like differentiating between apples and oranges.

They have similarities: The imaginary units of feet and meter helps us make sense of distance; and apples and oranges are fruits and are objects we can see. Just like when we count (quantify) the apples in the basket, when we measure something we are also quantifying.

Asking how many feet the height of the tree is just like when we answer this question how many oranges in the bag. We are visualizing how many feet can fit into the height of the tree, just like how many oranges can fit into the container. In the basket:

We counted six (6) apples.
We counted six (6) oranges. We can also say the quantity of oranges is 6.

We always ask “How many feet is the height of the tree?”. It is very similar to “How many eggs can fit into the bird’s nest?”, or “How many oranges can fit into the container?”.This also is based on ability to visualize the same object again and again when we measure something. This is just like how easy it is to visualize many apples in a container and how tight it feels when there are many people in a room.

We answer 6 feet. We mean we counted 6 feet. We can also say the quantity of feet is 6. We can also say 6 feet can fit into the height of the young banana tree.

Now you get why units of measurement like feet are called physical quantities. They can be counted like oranges and cats. But there is a semantic problem with the phrase “physical quantity” because feet is only an arbitrary point of reference to help us compare things.

So, semantically it is better to call them physical observables because we compare the size of objects based on what we have in our memory. Our five senses observe physical objects and we can imagine things with our unlimited creative potential.

Creating measuring devices

Imagining how many units or oranges can fit into a container is essential to creating measuring devices. The measuring device should be easy to carry, within our vision field and not very big compared to our body.

With all these characteristics in place we can define the length of the measuring device. If we compare the feet to the size of our body, we estimate it is 6 feet, using just our sense of sight and we remember how the unit of feel look like. We won’t create a measuring device longer than 6 feet.

Estimating the number is very similar to thinking of a random number, but the sense of sight helps, along with your memory, be more accurate with the randomness. You remember how the unit of foot was taught to you.

They go to the museum of units and take a small plank of wood with them. They place the small plank on top of the marking in the stone block and mark the two ends of the wood by scratching the wood with a screwdriver:

They can take home the unit of measurement called the “feet”. They can now take a bigger plank of wood that is slightly taller than a person and mark how many feet can fit into the bigger plank:

Marking 2 feet:

Marking 3 feet and so on:

When the feet is already duplicated to the other end of the large plank, write the sequence of HIndu-Arabic number symbols to the large plank as they count how many feet it is:

It now shows that 4 feet can fit in to the plank. Alternatively, just for show, the sequence of East Asian number symbols can be written in place of Hindu-Arabic symbols.

You discover the feet is convenient as a common comparison point because the amount of feet are already counted by writing the sequence of numbers in the plank. The decimal Hindu Arabic number writing system made it easy to compare with everyday objects. Since the numbers are already marked, it can do the counting for you.

Using measuring devices

So if you want to compare the length of any object, you place the larger plank of wood close to the object you want to compare it with, say a young banana tree that had just grown in your backyard.

You look from the bottom of the banana tree. You place the zero mark at the bottom of the tree to the starting point of the plank, meaning no feet. Then you look at the top of the trunk. You notice that the height is close to 3 feet, if you start from the bottom of the tree.

You place a basketball on the plank and compare it with the markings of the larger plank by putting one end of the basketball to the zero mark. You want to see how far the other end is from the zero mark. You saw it is 1 feet.

Now you feel you can compare any size of any object with the large plank of wood. You always place the zero mark to one end of the object. And then start counting from the “1” mark of the large plank.

So measuring is comparing and then counting with the Hindu-Arabic number writing system. The decimal display system is used along with our decimal vocabulary. Since we use a number writing system, it is a more convenient way to compare.

When we use a unit, comparing is also more convenient for our memories. Measuring cannot happen without counting from the zero mark of the measuring device and placing one end of the object to the zero mark. Counting stops at the other end of the object. The plank of wood became a measuring device because it was marked with numbers.

The word “Measure” is semantically convenient

Just as we imagine 5 apples a week are needed for a healthy digestion, we need a column that measures 10 feet for a house, just enough for a person’s height. The words “measure” and “compare” are semantically similar:

Measure the height of the banana tree.

That sentence is a shorter version of the phrase below. It can be written as:

Using your eyesight, compare the height of the banana tree with the unit of length called feet. Then count how many feet the banana tree is.

The word “measure” makes the second sentence more concise, but the disadvantage of that word, is that the origins of measurement is not obvious.

Method of definition of a unit is fully embedded to the method of measurement. We cannot define length without comparing based on a reference point. One of the things that define length is how we feel (sense of touch) about the size of the objects.

Measuring happens in two steps

1. Compare the object of interest with a reference point (usually units established by authorities) with an object from your memory. Other reference points can be a basketball or the size of your body.

2. Then count how many arbitrary reference points it takes (can fit in), with the object you want to measure. Example: “15 feet can fit in to the oak tree.” can be said as “the oak tree is 15 feet tall”

An authority has decided to use the feet because it is within our vision field. A lake or a mountain is beyond our vision field when we are near them. It also must be small enough with respect to our vision field.

The convenience of using units and measurement

All of the items in the rating system above will be compared with respect to a reference point. The reference point will be the unit called feet. The length or height of an object determines the size of the objects and can be compared to the feet.

When comparative vocabulary does not work any more, we effectively compare both the height of an oak tree and the height of a full-grown person to the feet. then from both the comparisons, we start counting them based on the feet. then after counting, we decide which is longer or shorter based on the what we have counted based on the defined sequence of numbers.

How many feet can fit in to the objects below?

Here are objects beyond the rating system. They may have been measured by a complicated mathematical formula because they massively exceed the size of our body and our measuring devices are just too small:

As we can see, using Hindu-Arabic numbers is more precise than using comparative vocabulary or rating systems because numbers let us know how close are the differences of the sizes. This one of the magic (usefulness) of numbers and counting.

Example: The difference between the height of the skyscraper (987 feet) and the mountain (380 feet) is much bigger (higher) than the difference between the height of the baby’s crib (4 feet) and the oak tree (9 feet).

Accuracy

Accuracy means that it must match the markings of the large plank of wood on where the feet was first defined. If it does not match the stone block, it is not accurate. Accuracy is often based on the material the measuring device was made. The large wooden plank we used throughout this serial is accurate:

Rubber planks are stretchable so they are not accurate for everyday use. The numbers and the markings on the rubber plank are also stretched, so it is not accurate with respect to the markings of on the stone block on the museum. Here is how the rubber plank looks like after stretching:

Precision

You find that the banana tree still has a small part that is shorter than a feet as shown by the big letter “I” symbol above, the measurement does not feel complete because of that small part. Something smaller than the feet can still fit in. Rating systems are not precise.

This is the same as the moment when you found out that, after putting the more important things in the baggage, it still has some small space left for some less important stuff you might like to bring with you. This is to the blue spaces remaining in the blue circle in Serial 2, after putting the small green circles.

To be precise in your measurement, you take a large piece of paper and mark the feet based on the stone block. You fold the piece of paper and then mark half a feet. Then fold it again. Then fold it again evenly by three. You call this the inch. Then the inch is duplicated in a similar way in a small plank as the foot. This is called subdividing the feet.

If you extra precision is needed for a certain situation: you come across an object smaller than an inch. When you notice a ladybug or an ant in your backyard, you recall the inch form your memory, they were smaller than that. The inch can be split into a smaller unit, just like the feet was split into a smaller unit called an inch.

You heard the idea of folding the foot to make the inch by watching the mass media and the conversion table you learnt from school. They just give the conversion table without explaining where the conversion came from. You start with a piece of paper that is foot-long.

You mark on the banana tree where it has measured 3 feet. You mark inches on the plank of wood. And at the start of the paper you count from where the plank left off (3 feet). You found the part that cannot be measured by feet measures 8 inches. You conclude that the height of the tree is 3 feet and eight inches:

Accuracy and precision in measurement

When you use a measuring device with most suitable material such as wood but not subdivided. That is accurate but not precise.

When you use a measuring device with the most suitable material then and subdivided into inches, that is precise and accurate. This is the most satisfactory tool for everyday use.

When you use a measuring device with not a suitable material such a rubber but not subdivided, that is imprecise and inaccurate, because it can stretch.

When you use a measuring device with not a suitable material such as a rubber but is subdivided, then that is precise but not accurate.

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