The Magic of Numbers – Serial 5

Ways of Displaying Numbers in the Hindu-Arabic Number writing system

Based on how many number characters are used in each place value, numerals e.g “nine thousand seven hundred and eighty two” can be displayed in writing differently from the usual “9782”.

In this article we will look at different ways to express numbers in written form based on the number character or symbols used in the slot machine or the number of beads in the abacus.

This article will demonstrate how the binary, octal and hexadecimal display systems show numbers. We will compare it to the everyday decimal display system used as a standard in the Hindu-Arabic number writing system.

Why did we introduce other number display systems?

Humans use fingers as their main medium for counting. The numeral vocabulary and number writing systems are based on the ten fingers, so the decimal display system is used everyday. The English word “decimal” is derived from the Latin word for ten.

Computers use on and off switches in their basic operation, so on and off switches are used as the medium of counting. Humans devised other number display systems to better build computers and anything related to computers (binary, octal and hexadecimal display systems).

1. Decimal Display System

Since we used the ten fingers in our hand as a tool for counting, this is the number display method we use in everyday measuring devices. Before liquid crystal displays were popular, slot machine-like displays on the measuring device were used for displaying readings. The popular example is the odometer in front of the driver’s seat of a car.

The decimal display system is sometimes called the base-ten system.

The number writing systems described in the previous section are decimal display systems (Hindu-Arabic, Roman, and East Asian). It uses ten characters in each place value (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). In abacuses, nine abacus beads can be put in each column, each column represents a place value. No beads in a column means zero (0).

In the previous two sections, we have demonstrated the decimal display system based on the slot machine and the abacus, but to compare it with the other display systems, we will take a look at it again:

We will use four slots in the slot machine illustrations to represent the first four place values. The old-style car odometer only has a limited amount of slots too. Starting from left, the 4 place value slots have names Thousands, Hundreds, Tens and Ones. The names are based on the decimal English numerals.

As we can see below, each of the 4 slots can hold the number characters 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, in a proper sequence.

This is how the sample number 3564 looks like in a slot machine and an abacus, respectively:

This is how the sample number 738 looks like in a slot machine and an abacus, respectively:

2. Binary Display System

Now imagine what if a car odometer uses the binary display system instead of the decimal display system. Everyone was taught the decimal display system since kindergarten or early elementary school without looking on the other display systems, so not everyone will be familiar with this.

This display system, along with octal and hexadecimal was only taught to college students wishing to be programmers, computer scientists and electrical engineers. The binary display system is sometimes called the base-two system.

We will use twelve slots in the slot machine illustrations to represent the first twelve place values. Unlike the decimal display system, the twelve place values slots don’t have names suited to the binary display system because there are no words yet from the English vocabulary to describe it.

The binary system displays 2 number characters (0 and 1) in each place value. Due to this word bit used in computers is derived from the terms binary and digit. Digit is another term for number character.

In abacuses, one abacus bead can be put in each column, each column represents a place value. No bead in a column means zero (0). Beads that are not used are put in the shaded area in the upper part of the abacus.

As we can see below, each of the 12 slots can hold the number characters 0 and 1. The decimal numbers 3,564 and 738 are displayed in binary below.

In a binary slot machine or an abacus, the decimal number 3,564 is displayed as 110111101100:

In a binary slot machine or an abacus, the decimal number 738 is displayed as 1011100010:

3. Octal Display System

It is derived from the binary display system. “Octal” is an English word derived from the Latin word for “eight”. The octal display system is sometimes called the base-eight system.

It uses 8 number characters (0, 1, 2, 3, 4, 5, 6 ,7) in each place value. Seven abacus beads can be put in each column, each column represents a place value. No beads in a column means zero (0).

We will use 4 slots in the slot machine illustrations to represent the first four place values. Unlike the decimal display system, the 4 place value slots don’t have names because there are no words yet from the English vocabulary to describe octal place values.

As we can see below, each of the 4 slots can hold the number characters 0, 1, 2, 3, 4, 5, 6, 7. The decimal numbers and are displayed in octal below:

In an octal slot machine or an abacus, the decimal number 3,564 is displayed as 6754:

In an octal slot machine or an abacus, the decimal number 738 is displayed as 1342:

4. Hexadecimal Display System

It is also derived from the binary display system. “Hexadecimal” is an English word derived from the Latin word for “sixteen”. The hexadecimal display system is sometimes called the base-sixteen system.

10 number characters (0, 1, 2, 3, 4, 5, 6, 7 ,8 ,9) are taken from the original decimal sequence of number characters and 6 (A, B, C, D, E, F) are taken from the sequence of first 6 capital letters of the Latin Alphabet when sung.

So altogether 16 number characters are used in each place value. 15 abacus beads can be put in each column, each column represents a place value. No beads in a column means zero (0).

But we can create all new number characters/symbols for the 6 characters taken from the Latin Alphabet, but the creator of the display system could be in a hurry to finish creating a new display system, so new symbols for them were not created. But this made it easy to type hexadecimal numbers in a computer keyboard.

As a result, large numbers in hexadecimal display will look like a nonsense mixture of numbers and letters to those who are familiar only with the decimal display system.

Example: The hexadecimal number CF5A7B is equivalent to 13,589,115 in the decimal display system. the hexadecimal number can be read as “thirteen million, five hundred eighty-nine thousand, one hundred and fifteen” in English numerals, which also is decimal.

We will use 4 slots in the slot machine illustrations to represent the first four place values. Unlike the decimal display system, the 4 place value slots don’t have names because there are no words yet from the English vocabulary to describe hexadecimal place values.

As we can see below, each of the 4 slots can hold the number characters 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, D, C, D, E, F. The decimal numbers 3564 and 738 are displayed in hexadecimal in the next page.

In a hexadecimal slot machine or an abacus, the decimal number 3,564 is displayed as DEC:

In an hexadecimal slot machine or an abacus, the decimal number 3,564 is displayed as 2E2:

Numerals, Language Vocabulary and Display systems

Below we will compare all of the number display systems and why there are not enough words from the English vocabulary for those displays systems other than the decimal display system.

To understand the binary, octal and hexadecimal counting systems, we rely on decimal counting system to describe those display systems, just as we are learning another language we need to rely on English or our mother tongue to understand the meanings of the foreign language. Therefore, converting a hexadecimal number to a decimal number is like translating a French word to an English word.

In this article we need to rely on the decimal display system to understand the binary, octal and hexadecimal display systems. We also need to rely on the decimal display system because English and most other languages have a decimal vocabulary, but no vocabulary to express the place values of binary, octal and hexadecimal.

The Maya language and other Mesoamerican cultures, have a vigesimal vocabulary and a vigesimal display system used in their number writing system. They have their own number symbols too. They have a size indication system similar to the Hindu-Arabic number writing system but displayed vertically instead of horizontally. Vigesimal means base-twenty. This display system won’t be discussed in detail.

A table showing the English and French numerals along with the decimal, binary, octal and hexadecimal display systems will be shown in the next page.

This is how the decimal numerals are integrated to our vocabulary: Hundreds mean ten tens. Thousand means ten hundreds. Million means thousand thousands. There is no equivalent vocabulary for the binary, octal and hexadecimal display systems:

The binary number “1111” is equivalent to “15” in the everyday decimal display system and “F” in the hexadecimal display system.

We cannot read aloud the binary number “1111” as: “one thousand one hundred and eleven”. The words, thousand, hundred and eleven are part of the decimal vocabulary we grew up with since kindergarten.

We also cannot read aloud the hexadecimal number 14 as “fourteen”. Again, fourteen is part of our decimal vocabulary. Since we don’t have a hexadecimal vocabulary yet, we have no choice but to call it “twenty” in English or “vingt” in French.

We have not developed a vocabulary that can help us read aloud the binary number so we have to stick with our decimal number vocabulary for a while: “1111” is read aloud as ” one eight, one four, one two and one one”. Or it can be read aloud as (eight plus four plus two plus one).

So you can now see that using binary, octal and hexadecimal numbers in everyday conversation is not reliable. Since we have no vocabulary from the English vocabulary to describe binary, octal and hexadecimal place values, we will use the decimal vocabulary as shown in the next pages.

Here are the first four place value names for the decimal display system:

Base-ten means as you go one place value to the left, the place value is ten times bigger than the one before it, i.e hundreds is ten times bigger than tens.

Being decimal, note that only the characters 0, 1, 2, 3, 4, 5, 6, 7, 9 can be displayed in each place value..This means maximum 9 abacus beads are allowed at each place value, no abacus at a column means 0.

The number 6524 is displayed as:

When an abacus is used:

Counting limit for a 4-column decimal abacus is 9999:

Here are the first eight place value names, in decimal vocabulary, for the binary display system:

Base-two means as you go one place value to the left, the place value is twice as big than the one before it, i.e “sixteen” is twice as big as “eight”.

Below, the decimal number 96 is displayed in binary, with the largest place value highlighted:

Being binary, note that only the characters 0 and 1 can be displayed in each place value. This means only one abacus bead are allowed at each place value, no abacus at a column means 0.

When a binary abacus is used:

Counting limit for an 8-column binary abacus is 11111111, which is 255 in decimal:



Here are the first four place value names, in decimal vocabulary, for the octal display system:

Base-eight means as you go one place value to the left, the place value is eight times as big than the one before it, i.e “sixty-fours” is eight times as big as “eights”.

Below, the decimal number 1354 is displayed in octal, with the largest place value highlighted:

Being octal, note that only the characters 0, 1, 2, 3, 4, 5, 6, 7 can be displayed in each place value. This means maximum seven abacus beads are allowed at each place value, no abacus at a column means 0.

When an octal abacus is used:

Counting limit for a 4-column octal abacus is, 7777 which is 4095 in decimal.:

Here are the first four place value names, in decimal vocabulary, for the hexadecimal display system:

Base-sixteen means as you go one place value to the left, the place value is sixteen times as big than the one before it, i.e “two-hundred and fifty sixes” is sixteen times as big as “sixteen”.

Below, the decimal number 7006 is displayed in hexadecimal, with the largest place value highlighted:

Being hexadecimal, note that only the characters 0, 1, 2, 3, 4, 5, 6, 7, 9, A, B, C, D, E, F can be displayed in each place value. This means maximum fifteen abacus beads are allowed at each place value, no abacus at a column means 0.

Note that reading aloud “B two hundred and fifty sixes” and “E ones” does not make sense compared to reading “one four-thousand-and-ninety-sixes” and “five sixteens” so we have to convert B and E from the table below:

The hexadecimal number B is 15 in decimal and E is 14 in decimal. We can now read aloud in decimal vocabulary “eleven two-hundred-and-fifty-sixes” and “fourteen ones”.

This is how the hexadecimal number 1B5E looks like in a hexadecimal abacus:

Counting limit for a 4-column hexadecimal abacus is FFFF, which is 65535 in decimal:

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