## Number System in Maths

**History of the Number System: How did the number system evolve?**

Numbers are an integral part of our daily lives, serving as the foundation for countless mathematical and scientific concepts. **But have you ever stopped to wonder how our ancestors developed the number systems we use today? **

**The Birth of Counting: Early Human Number Systems (History of the Number System initial stage)**

The development of the number system can be attributed to human efforts to arrange the concepts of size, form, and number during the Paleolithic era.

The earliest humans needed a way to count and keep track of quantities, leading to the development of rudimentary number systems. Paleolithic humans likely used simple tally marks, pebbles, or notches on bones and sticks to count objects. Over time, these primitive counting methods evolved into more sophisticated systems.

**Mesopotamia: The Cradle of Mathematics**

About 5000–6000 years ago, in Mesopotamia (modern-day Iraq), the first clear notations for numbers appeared **(one of the earliest known written number systems)**. Their system was based on a sexagesimal (base-60) system, which influenced subsequent civilizations in the region, including the Babylonians and Assyrians. These systems were pivotal in the development of mathematics, including geometry and algebra. Ancient number systems, which were an expansion of tally marks, included a variety of various symbols to denote higher numbers.

**The Egyptians: Hieroglyphic Numerals**

In ancient Egypt, hieroglyphic numerals emerged around 3000 BCE. The Egyptians used a decimal system, employing distinct symbols for powers of ten. Their numerical notation system was vital for various aspects of daily life, from trade and commerce to astronomy and engineering. The entire system of whole-number and positive rational number arithmetic was developed by the Egyptians and the Babylonians. **However, writing huge numbers required more effort with these methods.**

**Ancient India: The Birthplace of Zero and Decimal System**

One of the most significant contributions to mathematics came from ancient India. In the 6th century CE, Indian mathematicians introduced the concept of zero (0) and the decimal system (or base ten), revolutionizing mathematics and making complex calculations more accessible. The works of mathematicians like Brahmagupta and Aryabhata played a pivotal role in shaping our modern number system. Now, we could use just ten symbols to represent any number. Because it allowed for the representation of big numbers using fewer symbols, this system was much simpler to use than earlier systems. This approach led to the creation of the **Hindu-Arabic numeral system**, the foundation for the numbers we use today.

**The Arabic Influence: Transmission to the West**

During the Middle Ages, Islamic scholars played a crucial role in preserving and transmitting mathematical knowledge. They not only adopted Indian numerals (which we now call Arabic numerals) but also developed algebra and made significant advancements in trigonometry and arithmetic. This knowledge eventually reached Europe, sparking the Renaissance and the spread of the modern number system.

**The Western World: The Adoption of Arabic Numerals**

By the 10th century, Arabic numerals began to replace Roman numerals in Europe, primarily due to their efficiency and ease of use in calculations. This transition paved the way for the modern number system that we use today, where each digit’s position represents a power of ten.

**The Revolution of Calculus and Beyond**

The properties of the “real number system” (consisting of both rational and irrational numbers) began to be understood in the 1600s with the development of calculus, and by the end of the 1800s, mathematicians such as Dedekind and Cantor were giving rigorous mathematical definitions of the real number system. In the 17th century, luminaries like Isaac Newton and Gottfried Wilhelm Leibniz independently developed calculus, a branch of mathematics that revolutionized science and engineering. Calculus required a deep understanding of numbers and their relationships, driving further advancements in the field.

**Summary**

Overall, the number system has evolved over time, and the decimal system that we use today is the result of centuries of development and refinement. The history of number systems is a testament to humanity’s intellectual evolution. From simple tally marks to the sophisticated decimal system we use today, each civilization contributed to the development of numbers and mathematics. Our modern number system, with its Arabic numerals and place value, has unlocked countless scientific discoveries and technological innovations. As we move forward, it’s essential to remember and appreciate the rich history that brought us to this point, reminding us that mathematics is a universal language that transcends time and borders.

**What is Numerals?**

An abbreviation or term used to represent a number is called a numeral. It is a number-designating object, such as a symbol, group of symbols, or figure. An expression used to describe a numerical value is known as a number in linguistics. The traditional sign used to symbolize a number is known as a numeral. “5” or five is an example of a numeral that represents the number 5.

Numbers are symbols that humans use to represent ideas. A determiner that indicates the quantity of a noun is one way that numbers can be utilized. Words with different grammatical behavior fall under the category of “numeral” in English grammar.

**A number is expressed using digits, whilst a numeral can be expressed using words or symbols.**

**Numeral that is not a Number:**

An example of a numeral that is not a number is NaN, which stands for “Not a Number”. NaN is a value of a numeric data type that is undefined or un-representable, especially in floating-point arithmetic.

### What is radix in maths?

In mathematics, the term “radix” refers to the base or the number of unique digits used in a positional numeral system to represent numbers. The most common radix used in everyday mathematics is the decimal system, which has a radix of 10 because it uses ten unique digits (0 through 9) to represent all numbers.

However, radix can vary in different numeral systems. For example:

**Binary System (Radix 2):**In binary, the radix is 2 because it uses only two unique digits, 0 and 1.**Octal System (Radix 8):**In the octal system, the radix is 8 because it uses eight unique digits, 0 through 7.**Hexadecimal System (Radix 16):**In the hexadecimal system, the radix is 16 because it uses sixteen unique digits, 0 through 9 and A through F (representing values 10 through 15).

Different numeral systems are used in various contexts, with binary being common in computer science, octal and hexadecimal occasionally used in computing, and decimal being the most widely used in everyday mathematics and everyday life. The radix of a numeral system determines how numbers are represented and how place value works within that system.

**Where do numbers come from?**

The digits are combined to form a number. The quantities, measures, and locations that are represented by numbers are abstract mathematical concepts. They were developed so that people could measure and categorize their surroundings. The idea of numbers has changed over time, and many number systems have been created to meet varied requirements and objectives. **Here is a broad explanation of how numbers are made and comprehended:**

**Counting and Natural Numbers:**The most basic numbers are the natural numbers (also called counting numbers), which include 1, 2, 3, 4, and so on. These numbers are used for counting objects and are among the first mathematical concepts humans learn.**Whole Numbers:**Whole numbers include all the natural numbers along with zero (0). They are used for counting, ordering, and representing non-negative quantities.**Integers:**Integers are numbers that include all the whole numbers and their negative counterparts (e.g., -3, -2, -1, 0, 1, 2, 3). Integers are used for representing quantities that can be negative as well as positive, such as temperature changes.**Rational Numbers:**Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. For example, 1/2, -3/4, and 7 are rational numbers. They are used for representing ratios, proportions, and fractions of quantities.**Real Numbers:**Real numbers include all rational numbers and also irrational numbers. Irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimal expansions. Examples of real numbers include π (pi) and √2 (the square root of 2). Real numbers are used to represent quantities on the number line, including all possible values.**Complex Numbers:**Complex numbers are numbers in the form of a + bi, where “a” and “b” are real numbers, and “i” is the imaginary unit (√(-1)). Complex numbers are used in mathematics and engineering to solve equations that have no real solutions and have applications in electrical engineering, quantum mechanics, and other fields.

### **What is Place Value System?**

The place value system is a mathematical notation system used to represent numbers in a way that conveys the value of each digit based on its position or place in the numeral. In this system, the value of a digit depends not only on the digit itself but also on its position within the number. This is a fundamental concept in our modern decimal numeral system, which is also known as the base-10 place value system.

**Each digit has a fixed position called its place. **Each digit has a value depending on its place called the **place value of the digit**.

The **face value** of a digit for any place in the given number is the value of the digit itself.

**Place value of a digit = (face value of the digit) × (value of the place)**

A place value system is one in which the value of a digit in a number depends on where it appears in the number. Each location in a number in a place value system corresponds to a power of the base, which in the decimal system is 10. As you move from right to left, the places correspond to rising powers of 10:

- The rightmost position represents 10^0 (which is 1), so the digit in that position has a value equal to itself.
- The next position to the left represents 10^1 (which is 10), so the digit in that position is worth 10 times its face value.
- The next position represents 10^2 (which is 100), so the digit in that position is worth 100 times its face value.
- This pattern continues as you move further to the left, with each position representing a higher power of 10.

**For example:** In the number 352, the 5 is in the “tens” place, so its place value of 5 is 5* 10 = 50.

In the number 357:

- The digit 7 is in the rightmost position, so its value is 7 * 10^0 = 7.
- The digit 5 is in the middle position, so its value is 5 * 10^1 = 50.
- The digit 3 is in the leftmost position, so its value is 3 * 10^2 = 300.

When you add these values together, you get the total value of the number: 7 + 50 + 300 = 357.

Different numeral systems, such as binary (base-2), octal (base-8), and hexadecimal (base-16), also have their own place value systems based on powers of their respective bases.

**What is the Decimal Number System?**

The most widely used number system in the world is the decimal number system, sometimes called the base-10 number system. All numbers are represented by 10 distinct digits (0-9). A power of 10, which is the system’s foundation, is shown by the location of each digit in a decimal number. We can easily express and operate with numbers of different magnitudes thanks to this positional notation method. Decimal notation is the word used to describe the method of representing decimal numbers in base 10.

In the decimal system, each digit has a location and is 10 times more significant than the digit before it or to the right. For instance, since 25 is a decimal number, 2 is 10 times more than 5.

2025 = 2 x 10^3 + 0 x 10^2 + 2 x 10^1 + 5 x 10^0

35 = 3 x 10^1 + 5 x 10^0

The decimal system functions as follows:

**Digits:** There are 10 distinct digits in the decimal system: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

**Place Value:** The significance of each digit in a number is determined by its place in the number. Ones are represented by the top-right number, tens are by the second-to-left digit, hundreds are by the third, and so on. Each location represents a higher power of 10 in this pattern, which continues.

For example, in the number 3,372:

- The digit 2 is in the ones place, so its value is 2 * 10^0 = 2.
- The digit 7 is in the tens place, so its value is 7 * 10^1 = 70.
- The digit 3 is in the hundreds place, so its value is 3 * 10^2 = 300.
- The digit 3 is in the thousands place, so its value is 3 * 10^3 = 3000.

**Number Representation:** In the decimal system, every positive or negative whole number, decimal fraction, or mixed number may be represented by combining digits and their corresponding place values. For instance:

The number 124 stands for one hundred twenty-four.

Pi is the symbol for the mathematical constant 3.141592.

455 stands for four hundred fifty-five.

**Operations in Arithmetic:** The decimal system’s principles are used to perform operations in arithmetic, such as addition, subtraction, multiplication, and division. These procedures are simple and consistent because of the place value system.

**History of Indian Number System**

The introduction of 0 (zero) and the use of a decimal place value system are two of the most well-known features of the **Indian number system**. The brief history of the Indian number system is as follows:

**Origins in antiquity:**The Indus Valley Civilization, which flourished between 3300 and 1300 BCE in what is now Pakistan and northwest India, is where the Indian number system first appeared. These ancient tribes may have used some kind of numeral notation, according to archeological evidence.**Early Numerals:****Brahmi numerals**, which date back to the third century BCE, are the first known symbols for numbers used in India. Despite not being ideal for mathematical computations, these numbers were utilized for carvings and inscriptions.**Zero’s invention:**Around the fifth century CE, India established the idea of zero as both a placeholder and a numerical digit. The guidelines for using zero in mathematical operations were formalized by the Indian mathematician and astronomer Brahmagupta. Zero was a revolutionary idea since it made computations more effective and opened the door to the creation of a place.**Place Value System:**Around the 16 century CE, the decimal place value system, which is the basis of the contemporary**Indian number system**, started to take shape in India. In this method, a number’s value is depending on the position of each digit inside it, using powers of ten. Mathematical computations become considerably more versatile and efficient because to this method.**Transmission to the Islamic World and Europe:**Through commerce and cultural connections, the Islamic world received advancements in**Indian mathematics and number systems, such as zero and the decimal place value system**. Islamic scholars explored and refined these concepts further. Finally, these advances in mathematics made it to Europe, where they had a significant impact on the development of modern mathematics and the scientific revolution.**Adoption and Evolution:**Numerous civilizations all over the world eventually adopted and modified the Indian number system, which includes a decimal place value system and zero. The Indian numeric system was brought to Europe in the 13th century by the Italian mathematician Fibonacci (Leonardo of Pisa) in his work “Liber Abaci,” and it eventually acquired more recognition.

**Modern Number System**

The term “**Modern number system**” usually refers to the Hindu-Arabic numeral system, which is the most extensively used numeric system worldwide. The decimal system, as it is commonly called, is distinguished by the use of 10 distinct digits (0–9) and a positional notation scheme. India is where the current number system initially appeared, using the numerals 0 through 9 to represent various powers of 10. Eg., 0, 1, 2 ,3, 4, 5, 6, 7, 8, 9.

Overall, the development of science, technology, and civilization has been greatly aided by the contemporary number system, which has evolved into a crucial instrument for quantitative thinking and communication.