The Magic of Numbers – Serial 11

To understand that the word ‘time’ is mostly based on waiting and counting, we must imagine life without measuring devices or counting systems first, so we only have our five senses and our memory to make sense of sizes.

A scenario in the playground

Two mothers as friends, decided to take their children for a walk in the playground in Szczecin at sunshine. The two mothers were talking to each other about some everyday topics. The mothers did not bring any measuring instruments, such as clocks and watches with them, knowing it is a free day.

The children, one girl and one boy, played together by chasing each other. The two children want to show their mother who runs faster. The mothers stand up and walk to trees to observe the children. The mothers tell them that that the starting point is the pine tree and the end point is the mahogany tree. When they reach the mahogany tree, they have to stop running. The playground is illustrated below.

Not having stopwatches, the mothers have to wait for their children to reach the mahogany tree by looking at the children. But how can they accurately know who runs faster based on their eyes?

Both of the children run by starting from the pine tree, they start simultaneously. The mothers see that the girl reaches the mahogany tree before the boy can reach it. To the eyesight of the mothers, their children were moving because running is moving.

Who crosses the mahogany tree first is the fastest. Also the mothers know they covered the same distance/spacing because the trees did not move in their eyes. One mother has to wait a little longer for her boy to reach the mahogany tree. Waiting longer means slower with respect to the first child to reach the mahogany tree. So the girl is faster than the boy.

Our eyesight was built to detect moving objects and their trajectory as our survival instinct. Our eyes can tell the difference between fast and slow and not moving, so it can detect speed based on the movement of objects, without us thinking.

This is where our perception of time came from, the ability of our eyes to detect moving objects and their speed (how fast they are) and the ability of our ears to detect noise and wait for that noise to end. The scientific name for moving is to change position or “translation”.

In turn, we learnt to imagine things moving by visualizing it in our minds. We also visualize things stop moving.This is how expect waiting to finish. Based on this ability, we learnt to plan and set goals. How we conceived time in our imagination (minds) is our ability to see objects moving.

Our perception of time came from waiting.

The scientific name for waiting, or to wait is an interval. Just as the scientific name of the house cat is Felis catus. Throughout this serial we will use the the word interval.

You wait for the water to boil. You know from experience that the water has boiled if it briskly gives out bubbles. You know depending in the strength of the flame, you need to wait for the water to boil. So scientists say boiling water takes an interval. They mean you need to wait for the water to boil.

Waiting implies that we are expecting something to finish. So we observe water beginning to boil and we eventually observe it boiling. We might miss the water boiling if we don’t observe it. We define time as the interval of any event like boiling a water or the child running from the pine tree to mahogany tree.

In this scenario, we won’t use a clock. An interval is the how we wait for something to start and then finish. A child running from the palm tree to the mahogany tree is also an interval. Interval is a synonym of duration.

Then we remember we waited for the water to boil. That memory too, is called an interval. When we want to boil water next time, we remember the interval it takes to boil the water. We expect the interval to be the same with the same flame strength.

We know when something is removed from our senses, like noise, the noise has ended. We also use the word “interval” and “time taken” interchangeably. So we say the “time taken for the noise to stop is short” instead of the “interval of the noise is short”. We can also say ”We waited only shortly for the noise to stop”.

So intervals are memories that can be used as expectations. Intervals are experiences in our memory. Based on our memory of that interval and if that interval occurs again, we can expect that interval, from our memory, to behave similarly.

Rating intervals

We know everything takes an interval:

The train to stop at the next station.
Boiling water.
Doing housework.
Finishing a chapter in a book.
Planted seed to germinate.

We must at least experience them to know the interval. If the interval is too long, for example, waiting for a seed to germinate, we might do other things while waiting for it (wash the dishes).

From our experience we know the length of those intervals are different, so we many use comparative vocabulary to make sense of the many intervals we experience in our daily lives. Now that we have used comparative vocabulary we can use the star rating system with the first five symbols of Hindu-Arabic numbers to better compare any new intervals we experience.

We use the words long or short (absolute), comparative (longer or shorter) and superlative (longest and shortest), which are adjectives for describing intervals. We learnt these words from the English class.

Our first interval we experienced is the interval it takes to move our left arm from left to right or each step we take in crawling. The baby was aware it can move its arms and legs. Comparing it to the interval of moving the arms and legs:

We see that we can move our arms while the train is going to the next station, we rate is as one (1) star. We can call this interval short.

We remember that it took longer for the water to boil than to wait for the train to go to the next station, so we rate it as two (2) stars. We can call this interval quite short.

We remember that it took longer to do housework than to wait for the water to boil, so we rate it as three (3) stars. We can call this interval as medium.

We remember that it took longer than to finish a chapter in a book than to do housework, so we rate it as four (4) stars. We can call this quite long.

We remember that it took longer for a seed to germinate than the ones below, we can rate it as five (5) stars. We can call long.

So you say to yourself some superlatives: Waiting for a seed to germinate is the longest interval I have ever experienced and waiting for the train to arrive to the next station is the shortest interval I have ever experienced. All of the intervals above are compared to the interval taken for the person’s arm to move from left to right. The rating system is also illustrated below.

So the everyday vocabulary short, quite short, medium, quite long and long can be augmented with a star rating system. So when we rate any new interval, let’s say eating and drinking, we say it is quite short, so we rate it two stars, because we experienced that interval to be roughly as short as boiling water.

In the course of your life, you experience many intervals and compare the new ones based on your memory to make sense of the new intervals. The intervals in the list of reference ratings could be in your memory. This means any of the interval in your memory can be a reference point.

But what if you experienced an interval outside the rating system? Waiting for the planted seed to grow to a mature plant is beyond the rating system you have set up for yourself. Or you saw that it only took a blink of an eye to for an indicator light of any electronic device to flash.

You know it is difficult to remember various intervals. You also know all of those intervals are of different lengths and those intervals are irregular. Having the ability to plan, you start to look for regular intervals. An idea struck that it is good to use the number writing system and find a single reference point so that things are easier to compare.

In the next section we will take a look at regular and irregular intervals.

Irregular intervals

The intervals listed above are irregular intervals, because the length of the intervals are different and varied. We know each chapter in a book varies in length and some paragraphs may take some effort to be understood, so each time we finish a chapter in a book, the interval may be different length.

How do we know? At one time you start reading a chapter after you finished your lunch and finished it just before eating dinner. You need to pause for dinner for the next chapter. Based on that you called the interval of completing a chapter in a book as an irregular interval.

In counting, you have also experienced that reading aloud “one”, “two”, “three”, “four” and “five” takes a shorter interval than to read aloud “forty-two”, “three hundred and fifty three” or “one thousand five hundred and thirty-tree”. So you conclude counting is an irregular interval.

But since ancient times, when a race starts, the referee will always say, “At the count of three, get set go!”. You notice that the referee has to say “One, Two, Three”, only then the racers can start running. This means they have to use the counting system to let the racers get ready. Now imagine what will the racers feel if the referee counts to fifty. Numbers beyond ten take a longer interval to read aloud.

So you start to look for other moving objects to find regular intervals. The best thing you know is how the day and night progresses. Based on your sense of sight, day is when the sky is bright and night is when the sky is dark.

Observing regular intervals

Throughout the day, we observe that the sun is in the sky for the same interval as the day before, we call this morning and afternoon. And we notice that the dark sky also has the same interval when the sun is visible in the sky.

Almost all cultures observed that the sun “moves” as it day progresses from dawn to dusk. They noticed that the shadows of objects, such as trees, also move as the sun “moves” in the sky, an idea called the sundial was conceived. The cultures also know how to count based on the sequence of numerals, so they put number markings on the sundials as illustrated below:

To better keep track of its “movement” of the sun, sundials were invented. A sundial has a pointer in its center that will cast a shadow the plate. The plate is marked with the proper sequence of any cultural number writing system. The Roman number writing system is used above. The sequence of numbers helps in counting.

The farmers knew when it right to grow crops at the best season by looking at the surroundings, whether trees are blooming (spring), when the sun it at its hottest (summer), when leaves are falling (autumn), and when there is snow (winter).

They also observed the positions of the stars in the night sky. This is how astrology was conceived.

Using their native number writing system, those farmers counted how many days it took for each season to last and found that they are regular. Based on the counting, calendars were created for each season. They also saw that the pattern of “movement” of the moon in the evening has repeated three times and they call this the month. When the same season repeats, they say a year has passed.

Selecting a common comparison point for intervals

Why measure intervals? To know when to best do things like sleeping, doing household chores and the season to grow crops. We cannot measure without regular intervals as our common reference point. Selecting a comparison point or reference point is arbitrary, but should be easy for the eyes to see (must not be too big or too small)

First we need a regular interval instead of the irregular intervals listed in the rating system above and in counting. So we use the “movement” of the sun as our reference point. But we know reading a book takes shorter than the interval from dawn to dusk. So we need to look for a smaller reference point. We created the hours, minutes and seconds, which are ultimately based on the so called “movement” of the sun.

The ancient cultures used fractions, which is one of the key topics of mathematics. What is a fraction? When a whole is cut into two or more parts. Example: The ancients have to cut the whole tree into parts before a house can be made. So the definition of fraction is when the original whole is cut into smaller parts.

With the help of the knowledge of number and the native number writing systems, one culture cut a whole day into half to form the day and night. Then they cut the interval the sun is shining in the sky (day interval, from dawn to dusk) as a whole and cut it into twelve equal parts called hours. The number markings in the sundial are the hours.

They cut those whole hours into sixty equal parts to form a minute and they cut the whole minute into sixty equal parts to form the second. They they added those gradations (smaller ones) to the sundial plate with number symbols. This how they got more precision in knowing how far the sun has “moved”

So we have created smaller intervals called the hour, minute and second. These are based on the bigger interval called the day (day interval and night interval combined). Those smaller intervals can now supersede the rating system we have created earlier.

Using counting devices to keep track of intervals

To supersede the sundial, we eventually invented mechanical clocks based on the pendulum or electrical clocks to recreate the hours, minutes and the second. The clock has hands to count the hours, minutes and seconds. The same gradations are used for the minute and second and a different set of gradations are used for the hour.

As we all know telling the time has its own number display system. When shown in a slot machine, it works a bit differently from the decimal slot machine. In old-style information displays for airline fights, a mechanical slot machine-like display is used for Hindu-Arabic number symbols and the characters of the Latin Alphabet, instead of computer LCD monitors.

If you play with the clock application of your iPhone or iPod touch, and tap the timer tab, you will notice a slot machine based graphical user interface for number input.

You will notice that the hours slot only has 23 hours and the minutes slot only has 59 minutes. If we could add a seconds slot, it would only have 59 seconds. It cannot show 24 hours because that is counted as one day. It cannot show 60 minutes because that is counted as 1 hour. If a seconds slot was added, it also cannot show 60 seconds because that is counted as 1 minute. One of the picture shows that slot machine interface can be rotated by swiping the touch screen:

Selecting, for example, 8 hours and 42 minutes; and 5 hours:

How we measure intervals

We can now use our measuring instruments to compare the interval of the seconds with the intervals from the rating system above. From our experience in using the clocks, the irregular interval of boiling water will be compared with the regular interval of the minute in the example below.

We created stopwatches too to mark the start of the interval and the end of the interval. At the first boil we found that it took about 17 minutes or 1020 seconds, the next time it took 15 minutes or 900 seconds. You can also say 17 minutes can fit in to the interval of waiting for the water to boil.
So the stopwatch confirmed boiling water is an irregular interval. The stopwatch did the counting for us. Using the stopwatch we can compare the intervals listed in the rating system above to the minute or second.

The result of the stopwatch can help us plan when to boil water, if we have better things to do. So measuring is defined as comparing counting using an arbitrary reference point. The reference point can be called a “unit”. So the minute and second can be defined as units. The measuring device usually does the counting for us.

Every day we ask, “What is the time?”, instead we can ask: “How much has the clock counted, where are the positions of the clock hands?”, “What is the number in the clock now?”

Benefits of measuring intervals in planning

We can compare more precisely who runs faster in a race and take note of the best time to grow crops. Based on our ability to count, and our measuring devices for intervals, we can plan when to read the next chapter in the book.

The counting from the measuring device is stored in our memory. Then that memory can be used for planning events. An event is a series of intervals that have happened or planned/scheduled to happen. Those events can be called scenarios. This will be described in the next serial.

While we are doing something, sometimes we are so active that we want to know what to do next. This is why we plan. Also, our ability to expect things happening eventually led to the idea of planning, because we are conscious and creative beings.

So the counting systems helped very much in this aspect of our consciousness, that as stated above, we used counting systems to mark the best instant to begin the race, by saying “At the count of three, get set, go”. Clocks are derived from basic counting like: one, two, three…

Then based on counting systems, we created planning, this is similar to saying “At the count of one hundred and twenty, I will read a new chapter in the book.” Instead, we rely on the clock to do the counting for us, instead of counting aloud, because the minute or second is a regular interval (based on the interval from dawn to dusk). We will be wheezing by the time we say “one thousand three hundred and twenty four”.

Mechanical clocks can augment our body clock. Our body knows when to sleep and wake up. We know when to sleep if we are yawning. So we can record the position of the hands in the clock to know when to sleep after a busy day.

Our ability to perceive intervals may vary

Our ability to perceive intervals may vary with our mood and our age. Other species may perceive intervals more differently from us.

We experience that intervals may feel slow if we are excited about something to happen, for example we eagerly await our presents to be delivered in our house, or waiting for the next article to be posted at your favorite website.

When we are enjoying our company with friends, or when we enjoy what we do – when we look at the clock, time seems to fly.

Days seem to pass slower when we are children and days seem to pass by more quickly when we are already at an advanced age, this may be due to the speed of the neuronal networks changing as we age.

When houseflies view the television screen, they see 25 still images at the interval of a second like a picture slideshow, while we see smooth moving pictures. We all know from experience houseflies are very fast creatures, they go away when they immediately see our hand.

The counting we do on our clocks can be called linear time

Just as you marked the spacings between the cherry tree and the apple tree by drawing a line between them, you know it takes an interval for the child to run from the mahogany tree to the pine tree. You can visualize the children leaving a trail as they run. Based on our eyesight, we define lines as long and slender marks.

We leave footprints in the sand in the beach as we walk (move in the sands), we can visualize the footprints as a line from the bird’s eye view.

The children at the beginning of this article covered the spacing between the mahogany tree and the pine tree. We can picture the spacing the children have covered as similar to the line we have marked in the sticky note. This is why it can be called linear time.

In the course of the day, the sun casts a shadow on the sundial and due to the perceived “movement” of the sun. We all know when something moves, it covers a distance. So we perceive an hour, minute, and second as a movement. We too can visualize the shadow drawing a line as it moves in the number markings in the sundial. Since the shadow can move in the number markings, the sundial is doing the counting for us.

When we want to compare how fast an interval of something is (say the wind carrying away a maple leaf), we compare the trail of that moving object to the movements in the sundial. Just as the spacing of the pine tree and the mahogany tree did not change, the spacing between two numbers in the sundial does not change. Distances/spacings not changing is an important characteristic of regular intervals.

The maple leaf and shadow in the sundial both cover a distance. Since the movement of the maple leaf is an irregular interval while the movement of the shadow is regular, the sundial shadow is our common point of comparison. To understand linear time is the same as distance, we have to use a regular interval as a common reference point.

We can only know if a moving object is fast or slow if we compare the trail of the movement with a spacing that does not change, just as the movements of the boy and girl above. Running is an irregular interval. No clocks were used by the mothers, only their eyes, so they have to look for an common arbitrary reference point (spacings between trees).

Boiling water does no movement, but we know how fast or slow it is by looking at a counting device called the clock and how much it has counted. The clocks in the hand move like the shadow in the sundial, so they cover a distance.

The second in the clock or the hours in the sundial are regular intervals. When those intervals finishes, they create a trail with a regular distance. Counting based on a regular interval can help us determine which is fast and slow.

The clock did the counting for you. Irregular intervals cannot be counted. Regular intervals can. The predefined number sequence/continuum for counting is linear (1,2,3,4,5…), this is another reason it can be called linear time.
The question “How many times you close your fingers?” That question cannot be answered without knowledge of counting systems. The words “times” is attached with numbers each time we answer it: “7 times”.


Linear time is just a counting system (counting regular intervals) and it is similar to distance we measure with a ruler (spaces that don’t change). Regular intervals leave trails with distances/spaces that don’t change, as in the sundial and clock. The movement of an object can be a regular or an irregular interval.

Comparing any irregular interval (children running or boiling water) with regular intervals (day) or any arbitrary distance that does not change (spacing of the trees above) used as reference points is how the idea of linear time is created. A common reference point eventually becomes a universal reference point like the second, it is called universal because it used by everybody.

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