Types of Numbers in Math
Just like different individuals of the same family live in different homes, different numbers belong to the same family but have different types. Over time, various patterns of ten digits have been categorized into an array of number types. This framework of numbers varies from each other because of different properties and presentations.
That said, the types of numbers in maths are classified as per some purpose that they serve, property that they possess or fundamental rule that they follow.
What Are Numbers?
Those ten elegant digits, symbols, or numerals that we all learn early in life are the numbers. Numbers are algebraic in form and have a greatest influence in our lives in far more ways than we could ever think of.
Numbers in Real Life
Ever wondered what our lives would be like in absence of these 10 digits and the innumerable array of other numbers that they can create? Numbers are everywhere in our birth dates, ages, height, weight, addresses, phone numbers, credit card numbers, bank account numbers and a lot more.
Classification of Numbers
Numbers family can be classified in different categories. With that, we can also say that two or more types of numbers in maths can fall under one category. Refer to the image below for complete understanding of classification of numbers:
(Image to be added soon)
Different Types of Numbers
There are various types of numbers in maths. Let’s discuss some of the following:
Natural Numbers - the set of numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 that we see and use in our routine life. The natural numbers are commonly known as positive integers and counting numbers.
Whole Numbers - the natural numbers plus (+) the zero (0).
Real Numbers - set of real numbers in addition to all the rational and irrational numbers and are represented by the letter R. It also includes all the numbers that can be written in the decimal form.
Fractional Numbers - any number expressible by the quotient of two numbers as in m/m, "m" greater than 1, in which "m" is the numerator and "n" is the denominator.
Irrational Numbers - any number that is unable to be expressed by an integer or the ratio of two integers. These numbers are expressible only as decimal fractions in which the digits are ongoing with no repeating pattern. Examples of irrational numbers are √2 , √3.
Transcendental Numbers - any number unable to be the root of a polynomial equation with rational coefficients.
Quantum Numbers
Set of numbers used to define the energy and position of the electron in an atom are known as quantum numbers.
Types of Quantum Numbers
There are four quantum numbers that define the probable location of an electron in an atom which are as given:
The Azimuthal Quantum Number denoted by symbol ‘l’
The Magnetic Quantum Number denoted by symbol ‘ml’
The Principal Quantum Number denoted by symbol ‘n’
The Spin Projection Quantum Number denoted by symbol ‘ms’
Fun Facts
Almost all of us whether mathematicians, scientists, doctors, engineers, manufacturers, cashiers or carpenters could not survive without numbers.
Zero(0) as a number has the greatest value and importance.
Solved Examples
Example:
Find out the square root of -16? Write your answer in the form of imaginary number i.
Solution:
Step 1: Write the number in terms of square root √ (-16)
Step 2: Separate out -1. √ (16 × -1)
Step 3: Move apart square roots.√ (16) × √ (-1)
Step 4: Solve and simplify the square root. 4 × √ (-1)
Step 5: Write in mathematical terms of i.4i
Sometimes you get an imaginary solution to the equations.
Example 2
Simplify and solve the equation: a2 + 2 = 0
Step 1: Take the constant term on other side of the linear expression: a2 = -2
Step 2: Take the square root on both sides of the equation √a2 = +√-2 or -√-2
Step 3: Solve and simplify: a = √ (2) × √ (-1)
a = +√2i or -√2i
Step 4: Double check the answers by substituting values in the initial equation and see if we obtain 0. a2 + 2
(+√2i) 2 + 2 = -2 + 2 = 0 [since i = √-1 and square of i will be -1]
(-√2i) 2 + 2 = -2 + 2 = 0 [since i = √-1 and square of i will be -1]
Conclusion
we would not be able to live without numbers in our lives. Interestingly, there exists an almost infinite array of number types and hidden wonders emanating from these acquainted symbols that we use every day, the natural numbers.
FAQs on Number Types
1. What are the main types of numbers in Maths?
Mathematics classifies numbers into several types based on their properties. The main types are:
- Natural Numbers: Counting numbers starting from 1 (e.g., 1, 2, 3, ...).
- Whole Numbers: Natural numbers including zero (e.g., 0, 1, 2, ...).
- Integers: Whole numbers and their negative counterparts (e.g., -3, -2, -1, 0, 1, 2, 3, ...).
- Rational Numbers: Numbers that can be expressed as a fraction p/q, where p and q are integers and q is not zero (e.g., 1/2, -3/4, 5).
- Irrational Numbers: Numbers that cannot be expressed as a simple fraction, with non-terminating, non-repeating decimals (e.g., √2, π).
- Real Numbers: The set of all rational and irrational numbers.
- Complex Numbers: Numbers in the form a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit (√-1).
2. What is the key difference between rational and irrational numbers?
The key difference lies in their representation. A rational number can be written as a simple fraction (e.g., 0.5 = 1/2) and its decimal form either terminates or repeats. An irrational number cannot be written as a fraction of two integers, and its decimal representation goes on forever without repeating (e.g., π ≈ 3.14159...).
3. What are imaginary numbers and where are they used in the real world?
An imaginary number is a number that, when squared, gives a negative result. It is based on the imaginary unit 'i', defined as the square root of -1. While they don't measure physical quantities like real numbers, they have critical real-world applications in fields like:
- Electrical Engineering: To analyse alternating current (AC) circuits, voltage, and impedance.
- Signal Processing: In Fourier transforms for analysing sound waves, images, and other signals.
- Quantum Mechanics: To describe the wave functions of particles.
4. Why is zero considered a whole number but not a natural number?
This distinction is based on their fundamental purpose. Natural numbers {1, 2, 3, ...} are defined as 'counting numbers' used to count objects, which naturally starts from one. Whole numbers {0, 1, 2, 3, ...} extend this set to include the concept of 'nothing' or an absence of quantity, which is represented by zero. Therefore, zero is part of the whole number set but not the natural counting set.
5. How can you determine if a number is rational just by looking at its decimal form?
You can identify a rational number from its decimal form by checking for two conditions. The number is rational if its decimal representation either:
- Terminates (ends): For example, 0.25, 3.8, or 1.125.
- Repeats in a predictable pattern: For example, 0.333... (repeating 3) or 0.141414... (repeating 14).
If a decimal is non-terminating and non-repeating, it is an irrational number.
6. Are all integers considered rational numbers? Explain with an example.
Yes, all integers are also rational numbers. The definition of a rational number is any number that can be written in the form of a fraction p/q. Any integer can fit this definition by setting the denominator 'q' to 1. For example, the integer 7 can be written as the fraction 7/1, making it a rational number.
7. What is the importance of understanding different number types for school exams?
Understanding different number types is fundamental for success in mathematics as per the CBSE syllabus. Each type builds upon the last and is crucial for different topics. Integers are essential for algebra, rational numbers for fractions and ratios, and real numbers for calculus and geometry. Questions often test the properties and relationships between these types, such as identifying if a number is rational or irrational, which is a common topic in exams for Class 9 and 10.
8. What is the difference between 'number types' and 'number systems'?
While related, these are distinct concepts. Number types (e.g., real, integer, rational) classify numbers based on their mathematical characteristics and properties. In contrast, a number system is a method of writing or representing numbers. For example, the number 'ten' can be represented in the Decimal System (10), the Binary System (1010), or using Roman Numerals (X).
