Numeration System

The numeration system is a structure used to represent numbers and numerical concepts. Humans use numbers to express quantities, making it essential to understand the difference between numbers and digits. A number represents a quantity, while a digit is the symbol used to display that number.

Number and Digit

• Number: An abstract concept that represents a quantity, idea, or magnitude.
• Digit: A tangible symbol used to represent a number. For example, 5 is a digit that represents the number “five.”The symbols from 0 to 9, including “0” and “9”, are called digits. There will be no (minus) sign in front of the digit, if there is, it will not be a digit, the digit will stand alone. In other words, “-4, -8, -7” will not be a digit because there is a sign in front of it.

Number Sets

Numbers are grouped into sets based on their properties. These sets include:

• Natural Numbers: Consist of 0 and positive integers. They are commonly used for counting. (Example: 0, 1, 2, 3, …)
• Integers: Include natural numbers, zero, and negative integers. (Example: …, -2, -1, 0, 1, 2, …)
• Rational Numbers: Numbers that can be expressed as fractions. Any integer is also a rational number. (Example: 1/2, -3/4, 5)
• Irrational Numbers: Numbers that cannot be expressed as fractions. Examples include pi (π) and the square root of 2.
• Real Numbers: Include both rational and irrational numbers.

Number Systems and Their Uses

Number systems consist of symbols (digits) and rules used to represent numbers and perform mathematical operations. Various number systems were developed by different civilizations and regions to address their unique needs.

Hindu-Arabic Number System

The Hindu-Arabic Number System, which originated in India and spread to Europe, is the most commonly used system today. This decimal (base-10) system includes the digits 0, 1, 2, …, 9 and serves as a global standard in modern mathematics.

Example:

In the number 432:

• The digit 4 is in the hundreds place, representing 400.
• The digit 3 is in the tens place, representing 30.
• The digit 2 is in the ones place, representing 2.

Other Number Systems

Different number systems were developed throughout history to meet specific needs. These include:

• Tally Systems:
  Used by early humans, these systems employed physical objects, like stones or marks, to represent quantities. Each object or mark corresponded to a unit.
• Simple Grouping Systems:
  Numbers were represented in groups, such as in fives or tens. Commonly used in trade and record-keeping, this method is still seen in some traditional societies.
• Positional Systems:
  Foundational to modern mathematics, positional systems assign a value to a digit based on its position in the number. For example, in the base-10 system, a digit’s position determines whether it represents ones, tens, or hundreds.
• Arithmetic Systems:
  These systems define relationships between numbers through operations such as addition, subtraction, multiplication, and division, following specific rules.

Algorithms

An algorithm is a sequence of steps followed to solve a mathematical problem. Algorithms are not only the foundation of number systems but also fundamental building blocks in many scientific and technological disciplines.

The term algorithm is eponymous, deriving from Abu Abdullah Muhammad ibn Musa al-Khwarizmi. This scholar made a significant contribution to mathematics in the 9th century by documenting his algorithmic work in algebra.

As mentioned, algorithms are not limited to mathematics but are present in many aspects of our daily lives. For instance, when we want to cook, we follow recipes, which are step-by-step instructions for solving problems. Similarly, if we apply algorithmic instructions, we can even read.

In the modern world, algorithms are widely used not only in mathematics but also in computer science, engineering, and data analytics.

Number System Types

Numbers are not only organized in the base-10 system. Various number systems are used in daily life and specific fields. For example, the base-12 system is common in timekeeping, while the base-60 system is used for measuring minutes, seconds, and angles. Although the decimal system is predominant, binary and hexadecimal number systems are essential in digital electronics and microprocessor/microcontroller systems.

Decimal Number System (Base-10)

• Most widely used system.
• Digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
• Place Value: Each digit’s value is based on powers of 10.

Binary Number System (Base-2)

• Used in computer systems.
• Digits: 0 and 1.
• Place Value: Each digit’s value is based on powers of 2.

Octal Number System (Base-8)

• Digits: 0, 1, 2, 3, 4, 5, 6, 7.
• Applications: Commonly used in computer engineering and digital devices.

Hexadecimal Number System (Base-16)

• Digits: 0, 1, 2, …, 9, A, B, C, D, E, F (where A = 10 and F = 15).
• Applications: Used in computer memory addressing and color coding.

Conversions Between Number Systems

From Decimal to Another Base:

1. Divide the number repeatedly by the target base.
2. Record the remainders.
3. Write the remainders in reverse order to get the result.

From Another Base to Decimal:

Sum up all the results.

Multiply each digit by the base raised to the power of its position.

Source

Aksoy, Y. Kullandığımız Sayı Sistemleri.

Sobecki, David, and Allan Bluman. Math in Our World, McGraw-HillEducation