Numeral systems

Numerals, tally marks, in Egyptian hieroglyphics carved in a stone wall.

Numerals in Egyptian hieroglyphics. Temple of Karnak (16th - 11th century BCE).

"∩||" means 12. Can you figure out the meaning of some other numbers on this wall?

Photo credit: Peter van der Sluijs - Wikimedia Commons.

Tallying

The fundamental meaning of arithmetic is the art of counting. The word "arithmetic" is derived (via Latin) from the ancient Greek ἀριθμητική τέχνη (arithmētikḗ tékhnē), which literally means "art of counting". Arithmetic was, and still is, largely a verbal skill. However, eventually people needed a system to keep track of quantities such as number of property items or number of group members.

Between 50 000 and 10 000 years ago (the Late Stone Age) humans started to write numbers. They used tally marks by carving notches in bones or wooden sticks. Later, the need for more complex arithmetic with larger numbers in activities such as trade, accounting and time notation, led to the emergence of more advanced numeral systems. The early evolution of counting and writing numbers eventually led to the development of writing, money, measurement and science in general.

The most basic and simplest numeral system is a unary numeral system. A unary system basically comes down to counting or tallying, where every single count is marked by some symbol or numeral. Figure 2 shows an example.

Tally marks = 12

Fig.2. Tally marks. Counting strikes: 5 + 5 + 2 = 12.

Tally marks are typically clustered in groups of five for legibility. The orientation of the 5 strikes in a cluster used in Figure 2 is common in most of Europe and the English-speaking world. Other regions in the world may be used to other figures.

Roman numeral system

One slightly more advanced system than using tally marks is the numeral system represented by Roman numerals. Roman numerals were commonly used in the Roman Empire and in medieval Europe. They were gradually replaced by the decimal numeral system from the 14th century on. However, Roman numerals are still used in some specific cases.

Clock

Fig.3. Clock with Roman numerals.

What is the time?

Roman numerals are composed of letters from the Latin alphabet. Each letter (digit) represents a fixed given value:

Digit Value
I 1
V 5
X 10
L 50
C 100
D 500
M 1000

The basic idea is that if you want to determine the value of a number given in Roman numerals, simply add the values of all independent letters in that number.

IIII = 1 + 1 + 1 + 1 = 4

VII = 5 + 1 + 1 = 7

XX = 10 + 10 = 20

In "modern use" the letters are generally placed from left to right in order of value, starting with the largest. Exceptions to this rule indicate that values need to be subtracted. The exceptions are:

Some examples:

MDCCCIIII = MDCCCIV = 1804

MCMLIV = 1000 + (1000-100) + 50 + (5-1) = 1954

MCMXCIX = 1999

MMV = 2005

MMXVII = 2017

MMXX = 2020

What values do the next Roman numbers represent? Do you know an other, shorter, way to write them in Roman numerals?

MVV
MDCDLXLVIV

What values could the next Roman numbers represent, if we would use a slightly different convention?

MCMIC
MIM

Decimal numeral system

The decimal numeral system or Hindu–Arabic numeral system or base-ten numeral system is the most common numeral system in the world. It originated in India, about 2000 years ago. Via the Middle East, the system spread to Europe by the Late Middle Ages. From Europe it finally spread to the rest of the world.

Numerals 0 to 9 in old Indian script. With a bit of imagination one can recognize present day numerals in some of the old symbols

Fig.4. Digits used in the Bakhshali manuscript, presumably the oldest extant manuscript in Indian mathematics, dated to the early centuries of the Common Era.

From: "On the Bakshali manuscript" by August F.R. Hoernle, 1887.

The decimal numeral system has three distinctive properties:

The decimal system (decem meaning "ten" in Latin) uses precisely ten different symbols to compose any numeral. These symbols are called digits (digiti meaning "fingers" in Latin). Most used are the Hindu–Arabic digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Also Eastern Arabic digits and Devanagari (Indian) digits are used in the decimal system in various parts of the world.

The choice of ten digits is rather arbitrary, probably because we have ten fingers. Other examples of positional number systems are the binary (base-2) and hexadecimal (base-16) systems, both often used in computer technology.

The decimal numeral system is a positional or place-value system, which means that the value that a digit represents is depending on the position of this digit in the number. The Roman numeral system is, for instance, not a positional system; a V always represents the same value 5, regardless where it appears in a Roman number.

III = 1 + 1 + 1 = 3

111 ≠ 1 + 1 + 1

111 = 100 + 10 + 1

("≠" means "is not equal to")

Reading a decimal number from right to left:

  1. First the "ones position" or "units position", a digit times one,
  2. followed by the "tens position", a digit times ten (10),
  3. then the "hundreds position", a digit times hundred (100),
  4. then the "thousands position", a digit times thousand (1000),
  5. etc. etc.

A position in a decimal number represents a "power of ten" (1, 10, 100, 1000, etc.). The digit holding that position indicates how many of those particular powers of ten need to be included.

19023 = 3×1 + 2×10 + 0×100 + 9×1000 + 1×10 000
= 3 + 20 + 0 + 9000 + 10 000

Some examples, but now reading the given number from left to right, as we are used in our Latin script:

43 = 4×10 + 3×1 = 40 + 3

240 = 2×100 + 4×10 + 0×1 = 200 + 40 + 0

423 = 4×100 + 2×10 + 3×1 = 400 + 20 + 3

105 = 1×100 + 0×10 + 5×1 = 100 + 0 + 5 = 100 + 5

007 = 0 + 0 + 7×1 = 7

A zero at a position means: it is a void position. "Zero" represents "nothing".

While counting in base-10, after digit 9 in a position we start at digit 0 again for this position and add one digit to the next left position: ...,198, 199, 200, 201,...

Powers of ten

one:100 = 1
ten:101 = 10
hundred:102 = 100 = 10 × 10
thousand:103 = 1000 = 10 × 10 × 10 = 100 × 10
ten thousand:104 = 10 000 = 1000 × 10
hundred thousand:105 = 100 000 = 10 000 × 10 = 100 × 1000
million:106 = 1 000 000 = 100 000 × 10
ten million:107 = 10 000 000 = 1 000 000 × 10
hundred million:108 = 100 000 000 = 10 000 000 × 10

The naming of larger powers of ten, such as "milliard", "billion" and "trillion", is not consistent across languages. An American interprets "a billion" differently than a German. More about this in the next chapter.

An arithmetic operation like 103 is called exponentiation, which is repetitive multiplication. 24 means 2 × 2 × 2 × 2 = 16. Later mode about exponentiation.

The number 19023 is pronounced as "nineteen thousand (and) twenty three". "One thousand, nine hundred" (1900) can also be expressed as "nineteen hundred".

1900 = 1×1000 + 9×100, i.e. "one thousand, nine hundred".

1900 = 19×100, i.e. "nineteen hundred".

In 90 the 9 is in the tens position, in 120 the 12 is also in the tens position (120 = 12 × 10), in 1900 the 19 is in the hundreds position, hence 1900 = 19×100.

236 000 = 236×1000, i.e. "two hundred thirty six thousand".

3 000 000 = 3×1 000 000, i.e. "three million".

23 236 164 = 23×106 + 236×103 + 164×100, i.e. "twenty three million, two hundred thirty six thousand, one hundred sixty four".

Note the spaces used in large numbers, like 10 000. This space is the thousands separator. In European languages the digits in a large number are generally separated in groups of thousands, corresponding with the pronunciation of the number. Every three digits from the right, a "thousands separator" occurs.

Can you see why (10+2) × 10 = 100 + 20 ?

Hint:

120=12 × 10
=(10+2) × 10

What is in brackets belongs together as a whole. Add 10 and 2 first and then multiply by 10.

Can you see why 3 000 000 is written as 3×106 and 6 720 000 000 is written as 6.72×109 in "scientific notation"?