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Decimal Number System – With Types & Properties

Number System

A number is a mathematical value used for counting and measuring objects, and for performing arithmetic calculations. 

A number system or numeral system or system of numeration is a way of writing these numbers.  In other words, a number system is a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner.

Decimal Number System

There are many types of number systems in use today depending on their ease and compatibility with the applications they are used for. The most widely used is the Decimal Number System which uses $10$ symbols commonly referred to as digits to form the numbers.

Numbers in the decimal number system have various categories like natural numbers, whole numbers, integers, rational and irrational numbers, and real numbers.

The following figure shows the hierarchy of numbers in different categories of the decimal number system.

number system
Categories of numbers in a decimal number system

What is a Natural Number?

The natural numbers also known as counting numbers are a set of numbers starting from $1$ and moving on to $2$, $3$, $4$, and so on. These are called counting numbers because numbers are significantly used in our day-to-day activities and speech. We see these numbers everywhere around us, for counting objects, representing or exchanging money, measuring the temperature, telling the time, etc.

Natural numbers are represented by the letter ‘$N$’. $N = \{1, 2, 3, 4, ….\}$. There are infinite numbers in the set $N$, i.e., one cannot count all the natural numbers.

Since the natural numbers start from $1$, it is the smallest natural number and there is no largest natural number. You can find a natural number still larger than any large number by just adding $1$ to it.

The set of natural numbers can be shown on the number line below.

Number System
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Properties of Natural Numbers

The main properties are shown by natural numbers related to the four basic arithmetic operations, addition, subtraction, multiplication, and division. These properties are

  • Closure Property of Addition & Multiplication
  • Associative Property of Addition & Multiplication
  • Commutative Property of Addition & Multiplication
  • Distributive Property of Multiplication over Addition & Subtraction
  • Existence of Multiplicative Identity Property
Maths in Real Life

What is a Whole Number?

When a number $0$ is included in the set of natural numbers, it is called a set of whole numbers. As seen above the natural numbers are used as counting numbers. But many times you need to represent a number for ‘nothing’, that’s the reason symbol $0$ was introduced, and a set of natural numbers was extended to form a set of whole numbers.

Whole numbers are represented by the letter ‘$W$’. $W = \{0, 1, 2, 3, 4, ….\}$. There are infinite numbers in the set $W$, i.e., one cannot count all the whole numbers.

Since the whole numbers start from $0$, it is the smallest whole number and there is exist no largest whole number. You can find a whole number still larger than any large number by just adding $1$ to it.

The set of whole numbers can be shown on the number line below.

Number System

Properties of Whole Numbers

The main properties are shown by whole numbers related to the four basic arithmetic operations, addition, subtraction, multiplication, and division. These properties are

  • Closure Property of Addition & Multiplication
  • Associative Property of Addition & Multiplication
  • Commutative Property of Addition & Multiplication
  • Distributive Property of Multiplication over Addition & Subtraction
  • Existence of Additive Identity Property
  • Existence of Multiplicative Identity Property
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What is an Integer?

An integer is a signed number. By signed number, we mean a number associated with either a positive ‘$+$’ or a negative ‘$-$’ sign. 

A positive or negative sign before a number makes a huge difference. While positive implies add-on, negative implies decrease. With this idea, the concept of integers originated.

For example, if a profit of ₹$100$ is represented as $+100$ then a loss of ₹$100$ will be represented as $-100$. There are many other real-world applications where signed numbers or integers are used such as representing temperatures, an increase or a decrease in quantity, moving up or down, etc.

A set of integers is a set that includes negative and positive numbers, including zero. A set of integers, which is represented as $Z$, includes:

Positive Numbers: $1$, $2$, $3$, …

Negative Numbers: $-1$, $-2$, $-3$, …

Zero: defined as neither a negative number nor a positive number. It is a whole number.

$Z = \{… -4, -3, -2, -1, 0, 1, 2, 3, 4…\}$

The set of integers can be shown on the number line below.

Number System

Properties of Integers

The main properties are shown by integers related to the four basic arithmetic operations, addition, subtraction, multiplication, and division. These properties are

  • Closure Property of Addition, Subtraction & Multiplication
  • Associative Property of Addition & Multiplication
  • Commutative Property of Addition & Multiplication
  • Distributive Property of Multiplication over Addition & Subtraction
  • Existence of Additive Identity Property
  • Existence of Additive Inverse Property
  • Existence of Multiplicative Identity Property

What is a Rational Number?

The word ‘rational’ originated from the word ‘ratio’. So, rational numbers are the numbers related to the concept of fractions which represent ratios. If a number is expressed as a fraction where both the numerator and the denominator are integers, the number is called a rational number.

Mathematically, a rational number is represented as $\frac {p}{q}$, where both $p$ and $q$ belong to the set $Z$ (set of integers) and $q \ne 0$. The set of rational numbers is denoted by $Q$.

Note: If the denominator of a fraction is $0$, it becomes undefined, because dividing a number by $0$ is meaningless.

Rational numbers can also be expressed as numbers with a decimal point. There are two types of the decimal representation of rational numbers.

  • Terminating Decimal Numbers: are the numbers that do not terminate i.e., with uncountable decimal places, e.g., $2.5$, $-7.8$
  • Non-Terminating but Recurring Decimal Numbers: are the numbers that do not terminate but the decimal digits keep recurring (repeating), e.g., $3.222…$, $-5.434343…$

Note: Non-terminating and non-recurring decimal numbers are called irrational numbers.

Axioms, Postulate & Theorem

Properties of Rational Numbers

The main properties are shown by rational numbers related to the four basic arithmetic operations, addition, subtraction, multiplication, and division. These properties are

  • Closure Property of Addition, Subtraction, Multiplication & Division
  • Associative Property of Addition & Multiplication
  • Commutative Property of Addition & Multiplication
  • Distributive Property of Multiplication over Addition & Subtraction
  • Existence of Additive Identity Property
  • Existence of Additive Inverse Property
  • Existence of Multiplicative Identity Property
  • Existence of Multiplicative Inverse Property

What is an Irrational Number?

Irrational numbers are those real numbers that cannot be represented in the form of a ratio. In other words, those real numbers that are not rational numbers are known as irrational numbers. The set of irrational numbers is denoted by the letter $P$. It’s also denoted by $Q’$ or $\overline{Q}$, meaning ‘not $Q$’, i.e., not a rational number. 

Irrational numbers are the set of numbers that cannot be expressed in the form of a fraction, $\frac {p}{q}$ where $p$ and $q$ and $q \ne 0$. Also, the decimal expansion of an irrational number is neither terminating nor repeating.

Two of the most common examples of an irrational number are $\pi = 3⋅14159265…$ and Euler’s constant $e = 2.7182818…..$. The other examples are $\sqrt{2}$, $\sqrt[3]{2}$, $\sqrt[3]{3}$, $\sqrt[4]{5}$, $\sqrt[7]{11}$.

Properties of Irrational Numbers

Given below are some of the properties of irrational numbers:

  • Closure Property of Addition and Subtraction
  • Associative Property of Addition & Multiplication
  • Commutative Property of Addition & Multiplication
  • Distributive Property of Multiplication over Addition & Subtraction
  • Existence of Additive Inverse Property
  • Existence of Multiplicative Inverse Property
  • Irrational numbers consist of non-terminating and non-recurring decimals

What are Real Numbers?

Any number that can be found in the real world is a real number. The set of real numbers consists of all the natural numbers, whole numbers,  integers, rational numbers, and irrational numbers. In fact, the set of real numbers denoted by $R$ is a superset of all the other sets of numbers, i.e., $R$ = $N + W + Z + Q + P$.

Properties of Real Numbers

Given below are some of the properties of irrational numbers:

  • Closure Property of Addition, Subtraction, Multiplication & Division
  • Associative Property of Addition & Multiplication
  • Commutative Property of Addition & Multiplication
  • Distributive Property of Multiplication over Addition & Subtraction
  • Existence of Additive Inverse Property
  • Existence of Additive Identity Property
  • Existence of Multiplicative Inverse Property
  • Existence of Multiplicative Identity Property

Conclusion

We use numbers in our daily lives. Depending on the application and situation, the numbers used are of different types. The broad categories of numbers used by us are – Natural Numbers, Whole Numbers, Integers, Rational Numbers, and Irrational Numbers. The set of Real Numbers consists of all the other categories.

Recommended Reading

FAQs

What is Decimal Number System?

A decimal number system uses $10$ symbols commonly known as digits. The digits used in decimal number system are $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $0$.

Where is Decimal Number System used?

We use the decimal number system in our day-to-day life, whether it is money, measurements like length/distance, weights, or capacities.

What is Decimal Number System also called?

The decimal number system is also called a base $10$ number system since it uses $10$ symbols or digits. All the numbers in the decimal number system are formed from the digits $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $0$.

What are the four important properties of decimal numbers?

The four important properties of decimal numbers are commutative property, associative property, distributive property, and identity property.

What are different types of decimal numbers?

There are many different categories of decimal numbers. But broadly decimal numbers are categorized as natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.

Which of the sets – natural numbers and integers is smaller?

The set of natural numbers is smaller compared to the set of integers. The hierarchy of different sets of decimal numbers from highest to lowest is real numbers ( = rational numbers + irrational numbers) > integers > whole numbers > natural numbers.

Is $0$ present in all the different sets of decimal numbers?

$0$ is present in all the different sets of decimal numbers except the set of natural numbers.

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