Numbers are fundamental entities that play a crucial role in mathematics and the understanding of the world around us. They represent quantities, magnitudes, and measurements and are essential for counting, calculating, and solving problems in various fields. Different types of numbers exist, each with unique characteristics and applications.
Natural Numbers: Natural numbers are the most basic type of numbers, representing counting numbers that start from one and continue indefinitely. They are used for simple counting purposes and form the building blocks for more complex number systems.
Whole Numbers: Whole numbers include zero along with all the natural numbers. They are useful for representing quantities that cannot be negative, such as the number of objects or elements in a set.
Integers: Integers encompass all positive and negative whole numbers, including zero. They are fundamental in algebraic operations and are used in various mathematical models and real-world scenarios.
Rational Numbers: Rational numbers can be expressed as fractions, where the numerator and denominator are integers and the denominator is not zero. They find application in proportions, ratios, and measurements.
Irrational Numbers: Irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimal expansions. They add a level of complexity to mathematical calculations and often emerge in geometric and trigonometric contexts.
Real Numbers: Real numbers comprise both rational and irrational numbers, forming a continuous number line. They are vital for representing any quantity, from the tiniest fraction to the most significant decimal.
Imaginary Numbers: Imaginary numbers are introduced to solve certain mathematical problems and are represented as multiples of the imaginary unit "i," where "i" is the square root of -1.
Complex Numbers: Complex numbers combine real and imaginary numbers, expressed in the form a + bi. They are used extensively in physics, engineering, and other scientific fields to describe quantities with both magnitude and direction.
The importance of numbers lies in their ubiquity and universality across disciplines. They serve as a fundamental tool for scientific research, technological advancements, and everyday life. From basic arithmetic to advanced calculus, numbers are the language of science and the basis for making informed decisions. They underpin financial transactions, allow us to measure and quantify physical phenomena, and enable us to understand abstract concepts in mathematics. Moreover, the various types of numbers enrich our understanding of mathematical structures and relationships, empowering us to analyze patterns, solve problems, and make discoveries that shape the world we live in. Whether in science, engineering, economics, or any other domain, numbers hold an indispensable position in human knowledge and progress.
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• Numbers are the foundation of mathematics and play a
crucial role in various fields.
• Mathematics is built on numbers which also pervade every
area of our life.
• Numbers are fundamental to our comprehension of the
world.
• In this presentation we will explore different types of numbers
and their properties.
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• Counting numbers that start from 1 and
continue indefinitely.
• They are represented as 1, 2, 3, 4, 5, …….
• They are used for simple counting and basic
arithmetic operations.
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• All natural numbers including zero constitute whole numbers.
• In other words, counting numbers starting from 0 are called
whole numbers.
• They are denoted as 0, 1, 2, 3, 4, 5……
• They are useful in computer programming and counting
objects.
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• They comprise all positive and negative whole numbers
including zero.
• The set of integers is denoted by ….., -3, -2, -1, 0, 1, 2, 3,….
• Integers play an important role in algebraic operations and
mathematical modelling.
• All positive and negative numbers including 0 comprise
integers.
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• These numbers can be expressed as fractions, where
numerator and denominator are integers and denominator is
not zero.
• In other words, rational number is a number which can be
expressed in P/Q form, where p and q are integers and q is
not equal to zero.
• Eg: 1/2, -3/4, 0, 7 etc.
• Include positive, negative, and zero values.
• Widely used in ratios and proportions.
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• Cannot be expressed as fractions and have
non-repeating, non-terminating decimal
expansions.
• Eg: π (pi), √2, 5.099019513….. etc.
• They have fascinating properties and are
commonly encountered in geometry and
trigonometry.
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• Real numbers encompass both rational and irrational
numbers.
• They form a continuous number line with every possible
value.
• Widely used in scientific calculations and measurements.
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• They are represented as multiple of imaginary unit “I”, where “I” is
the square root of -1.
• Let us take an example: 5i, Where, 5 is the real number and I is the
imaginary unit. When this number 5i is squared, we will get the
negative result as -25. This is because the value of i2 is -1. This
means that the √-1=i.
• The notation “I” is the foundation of all the imaginary numbers. The
solution written by using this imaginary numbers in the form a+bi is
known as complex numbers.
• Used to solve certain mathematical problems and are essential in
complex number operations.
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• Combination of real and imaginary numbers.
• Expressed in the form a+bi, where “a” and ”b” are real
numbers, and “I” is the imaginary unit.
• Widely used in engineering, physics and electrical circuits.
• Eg: 3+4i, 7-13.6i, 2+I, 0+25i=25i
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• Prime numbers are the natural numbers greater than 1 that
have no positive divisors other than 1 and themselves.
• Eg: 2, 3, 5, 7, 11, 13, etc.
• They are fundamental in number theory and have
applications in cryptography and computer science.
• Cryptography:Method of protecting information and
communications through the use of codes.
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• Composite numbers are natural numbers
greater than 1 that have more than two positive
divisors.
• They can be formed by multiplying two or more
prime numbers together.
• Eg: 4, 6, 8, 9, 10, 12, etc.
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• Numbers can be classified as positive, negative, or zero.
• Positive numbers are greater than zero while negative
numbers are less than zero, and zero is neither positive nor
negative.
• These concepts are essential in repeating quantities with
direction and magnitude, such as temperature, profit, loss
etc.
• Eg: 1, 2, 3, 4, -1, -2, -3, -4, etc.
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• Natural numbers can be categorised as odd or even.
• Odd numbers are integers that are nor divisible by 2. Eg: 1,
3 , 5, 7, etc.
• Even numbers are divisible by 2. Eg: 2, 4, 6, 8, etc.
• Even numbers find applications in various real-world
scenarios, such as arranging items in pairs or groups.
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• Transcendental numbers are a subset of real numbers that
are neither algebraic nor rational.
• They cannot be the solution to any non-zero polynomial
equation with integer coefficients.
• Well-known examples: π (pi) and e (Euler’s number).
• Transcendental numbers have indefinite decimal expansions
and do not follow any pattern.
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• Prime factorisation involves expressing a composite number
as a product of prime factors.
• It is a fundamental concept in number theory and is used in
various mathematical algorithms.
• For eg: prime factorisation of 24= 2☓2☓2☓3= 23☓3
• FUNDAMENTAL THEOREM OF ARITHMETIC: Every
composite number can be expressed (factorised) as a
product of primes, and this factorisation is unique, apart from
the order in which the prime factors occur.
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• Numbers come in various types, each with unique properties and
applications.
• Understanding these types of numbers enriches our mathematical
knowledge and problem solving abilities.
• From natural numbers to complex numbers, numbers play a fundamental
role in shaping our understanding of the world.
• Numbers are not merely abstract concepts but have profound implications
in diverse fields of science, technology and everyday life.
• Continuously exploring the world of numbers unlocks new possibilities and
opportunities for innovation and discovery.
