The Magic of Numbers – Serial 3

Determining the size of a large number

How do we know if a number is big? We use the place value system.

We see a pattern in the combined numerals: The larger order numerals are leftmost and smaller numerals are rightmost. The size of the number is indicated by how many zeros behind the left most number, as in three hundred million.

Just as the word “cat” has 3 (three) letters, “forty two thousand, two hundred forty” six has 5 (five) symbols. The symbols called numbers can also be called “number characters”.

So we can say that the number 42,246 has five (5) number characters. Commas (,) are sometimes used in numbers to make the number characters easier to see. The more number characters used in a number from end to end, the larger the number is, as shown in the table above.

There are altogether ten (10) number characters: ”0,1,2,3,4,5,6,7,8,9”. The other word for number characters is “digit”. The word digit is also another word for finger and toe. This can suggest that counting began with fingers.

Similarly we have altogether 26 (twenty-six) Latin alphabet characters, often called “letters”. Those “letters” are symbols of the sounds we can create with our voice box.

In number writing systems, we use place value to indicate the size of the numbers we write. The number writing system we have been taught since kindergarten is the Hindu-Arabic number writing system.

A number writing system will always consist of a set of symbols and a number size indication system to indicate size. The number symbols we have used so far in this article is called Hindu-Arabic number characters/symbols.

The number size indication system in the Hindu-Arabic number writing system is called the place value system.

The more number characters are used in a number, the higher the place value of the number.

Example: Some people might say the number 42,246 has “five (5) digits”, so the word digit has become confused with “place value”. So it is better to call it “a five (5) place value number”, to avoid semantic confusion. Also it is accurate to call it “five (5) number character number”. The word “digit” is less precisely defined than “number character”.

Before the place value system became popular, the abacus was used because reading and writing was not popular in the past.

The place value system is also called the positional notation system, the position of a number character in the number indicates its size:

In the number “1678”, the position of 7 is in the Tens. The position of 6 in the Hundreds. Hundreds and tens are size indicators, thus they are positions.

The Hindu-Arabic Place value system

We will take detailed look at how the Hindu-Arabic place value system works on the infographic in the next pages. Simply put, it works like a slot machine or an abacus.

There is a place for millions, thousands, hundreds, tens and ones. Ones can also be called “Units” but there will be semantic clash with the word “Units” in measurement systems, so in this article, we will use “Ones” in place value systems.

Place value follows a proper sequence, just as there is a proper sequence of numerals needed to make sense of counting. Based on the ten fingers we use in counting, as you go one place to the left, the size will increase ten times. Example: Thousands is ten times higher than Hundreds.

Any number, except zeros on the leftmost place determines the size of the number. Each place can only use the 10 (ten) number characters, one of them zero (0). The proper sequence is shown below.

The sequence is improper below, if it is used nobody will understand you because it is not logical based on the definition of the place value names, to determine size:

If we include the millions, we have these places in the example number below, highest value highlighted:

We can also omit the higher places if we want to use smaller order numbers:

If we want to include the higher order place values, we can add zeros to the left of the number but in practical use it is not necessary and is redundant. Because the highest place value that has numbers is enough to indicate the size (highlighted). It simply indicates those places have nothing:

Below, since it is the numeral “hundreds” that indicates the size of the number, the zeros (0) on the right must be added to indicate place value. Hundreds must always show three (3) number characters. It also indicates there no tens and ones but only hundreds. The zeros (0) on the left can be added but it is redundant.

In the number below, there are no thousand and tens, but the number must always show five (5) number characters, because its highest position is the Ten Thousands:

If there is nothing in all places, it is best to write “0” rather than “0000” to indicate nothing.

Abbreviating the place value labels we have, highest place value highlighted:

The highest place value is highlighted (hundreds, H). It can be written in text as “537”.

How does the place value system work like a slot machine?

Since counting or numbers is infinite, the place values that can be shown in the previous section are also infinite. We use the slot machine to show how many number characters can be used in each place value. Before car odometers used liquid crystal displays (LCD), they used a mechanical odometer that works like almost like a slot machine.

To demonstrate how the place value system works like a slot machine in this chapter, we will use only the first 4 place values:

They are abbreviated as:

This is how the slot machine shows zero inside the black rectangle:

Start counting at 1, we turn the slot up at ones place value slot:

Turning the slot indicates increasing and turning down the slot indicates decreasing.  We turn the wheel up as we count to 6:

Eventually for the Ones place value we can only turn it up until 9:

To continue counting, the next number (10) is shown by resetting the ones place value slot to 0 (to indicate no ones) and then lifting the tens place value slot to 1:

Counting to 16 will show this, indicating 1 ten and 6 ones:

Reaching 28 (2 tens and 8 ones):

Reaching 39 (3 tens and 9 ones):

To count to 40 we reset the ones place value slot to 0 and lift the tens place value up one turn:

We eventually reach 99 (9 tens and 9 ones):

To show 100, the hundreds place value is lifted to 1 and then the tens and ones place value slot is reset to 0, indicating no tens and ones but only hundreds.

Eventually we reach the counting limit of the 4-place value slot machine, so it will show the numeral “nine thousand nine hundred and ninety-nine”:

How does the place value system work like an abacus?

Here, we will use the abacus to show how many number characters can be used in each place value. Abacuses were used in many cultures before the electronic calculator or computer was invented. In the olden days it was a useful and practical tool for calculations.

To demonstrate how the place value system works like an abacus in this chapter, we will use only the first 4 place values:

The abbreviated place values are shown at the bottom of the abacus:

This is how the abacus shows zero, when all place values are empty. Beads that are not used are shown on the upper part (shaded) of the abacus.

Start counting at 1, we place one bead in the ones slot:

Adding more beads to the lower part indicates increasing and removing beads from the lower part indicates decreasing. We put more beads as we count to 6:

Eventually for the Ones place value we reach the maximum amount of beads, showing nine:

To continue counting, the next number (10) is shown by putting all of the beads of the ones place value to the shaded area to indicate 0 (to indicate no ones) and then adding one bead to tens place value slot to show 1 ten:

Counting to 16 will show this, indicating 1 ten and 6 ones:

Reaching 28 (2 tens and 8 ones):

Reaching 39 (3 tens and 9 ones):

To count to 40 we put all the beads of the ones place value to the upper part to show 0 ones and put one more bead to the tens place value:

We eventually reach 99 (9 tens and 9 ones):

To show 100, one bead is added to the hundreds column and then the beads are removed from the tens and ones place value, indicating no tens and ones.

Eventually we will reach the counting limit of the 4-place value abacus, so it will show the numeral “nine thousand nine hundred and ninety-nine”:

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