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Fundamental Theorem of Arithmetic
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Today weβre going to discuss the Fundamental Theorem of Arithmetic, which states that every composite number can be expressed uniquely as a product of primes. Can anyone give me an example of a composite number?
How about 12? It can be factorized into 2 and 3.
Exactly! Now, 12 can be factored as 2 Γ 2 Γ 3 or in exponential form, 2Β² Γ 3. This uniqueness is important. Remember: if you rearrange the multiplication, you still arrive at the same number.
Does that mean whatever we multiply, as long as they are primes, we can always get a composite number?
Yes! Thatβs right! In fact, there are infinitely many primes and combinations of them yield infinitely many composite numbers.
Letβs recall the acronym βFATβ which stands for **F**actorization of numbers into primes being **A**lways **T**ransparent. This helps us remember the essence of the theorem.
So every composite number has a unique prime factorization?
Correct! This is crucial for finding HCF and LCM which we will explore in depth as we continue.
In summary, the Fundamental Theorem of Arithmetic tells us about the unique prime factorization of numbers, greatly aiding our understanding of integers.
Proving Irrational Numbers
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Now let's turn to irrational numbers. Who can tell me what an irrational number is?
I know itβs a number that canβt be written as a simple fraction.
Exactly! Now, we are going to prove that 2 is irrational. Weβll use a method called 'proof by contradiction.' First, we assume 2 can be expressed as a fraction. How can we represent that?
We can say itβs like r/s, where r and s are integers.
Correct! Now suppose r and s have a common factor. If we square both sides, what do we find?
We find that 2bΒ² = aΒ², which means 2 divides aΒ², so 2 must divide a.
Right! This leads to a contradiction because it implies both a and b are even, contradicting our initial condition of them being coprime. Thus, we conclude that 2 is irrational.
Can we apply that same logic to prove that 3 is irrational?
Absolutely! The approach is nearly identical. Remember the steps we just discussed!
To summarize, we showed that by assuming an integer can be represented as a fraction, we reached a contradiction, proving itβs irrational.
Applications of Unique Factorization
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Letβs discuss how we can apply the Fundamental Theorem of Arithmetic to find HCF and LCM of given numbers.
Can you give us an example?
Sure! Letβs find the HCF and LCM of 6 and 20. First, what are the prime factorizations of these numbers?
I think 6 is 2 Γ 3 and 20 is 2 Γ 2 Γ 5!
Correct! For HCF, we take the smallest power of each common prime. What does that give us?
So, HCF is 2. And for LCM, we take the highest powers!
Right again! Thus, LCM is 60. Can anyone verify the relationship between HCF and LCM?
HCF multiplied by LCM should equal the product of both numbers.
Yes! Keep that in mind as itβs a very useful relationship! Remember: 'HCF and LCM make products easy!'
In summary, we explored prime factorizationβs role in calculating HCF and LCM, showcasing its practical applications.
Introduction & Overview
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Quick Overview
Standard
In this section, we delve into the significance of the Fundamental Theorem of Arithmetic regarding the unique factorization of composite numbers into primes. We also begin to prove the irrationality of specific numbers such as 2 and 3, setting the foundational understanding for real numbers in mathematics.
Detailed
Detailed Summary of Real Numbers
In this section, we begin by revisiting the world of real numbers, particularly focusing on their mathematical properties. The Fundamental Theorem of Arithmetic states that every composite number can be expressed uniquely as a product of prime numbers, emphasizing that this factorization is a cornerstone of number theory. This theorem not only helps in recognizing the properties of numbers but also assists in practical applications such as finding the Highest Common Factor (HCF) and the Least Common Multiple (LCM) using prime factorizations.
Furthermore, we explore the implications of this theorem through examples and exercises, highlighting its utility in establishing the irrationality of certain numbers. The section proceeds to prove the irrationality of 2 and 3 through contradiction, showcasing the deeper aspects of number properties, including how they connect to the definition of real numbers. Understanding these foundations is crucial as they pave the way for further studies in irrational numbers and their representations.
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Introduction to Real Numbers
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In Class IX, you began your exploration of the world of real numbers and encountered irrational numbers. We continue our discussion on real numbers in this chapter. We begin with two very important properties of positive integers in Sections 1.2 and 1.3, namely the Euclidβs division algorithm and the Fundamental Theorem of Arithmetic.
Detailed Explanation
This chunk introduces the concept of real numbers and builds a link to previous learning about irrational numbers. Students will recall that real numbers include both rational and irrational numbers. The section promises a deeper exploration of two significant mathematical properties: Euclidβs division algorithm (which relates to integer division) and the Fundamental Theorem of Arithmetic (which concerns the uniqueness of prime factorization).
Examples & Analogies
Consider a baker who needs to divide 10 pies among 3 friends. In doing this, they may find that each friend can have 3 whole pies, and 1 pie remains. This is similar to Euclid's division algorithm, where the pie division results in a quotient (3) and a remainder (1).
Euclidβs Division Algorithm
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Euclidβs division algorithm, as the name suggests, has to do with divisibility of integers. Stated simply, it says any positive integer a can be divided by another positive integer b in such a way that it leaves a remainder r that is smaller than b. Many of you probably recognise this as the usual long division process.
Detailed Explanation
The algorithm states that for any two positive integers, 'a' and 'b', we can express 'a' in terms of 'b' as follows: a = b*q + r, where 'q' is the quotient (the result of division without decimals) and 'r' is the remainder (the leftover amount). The condition is that 'r' must be smaller than 'b'. This concept is foundational for understanding operations with integers.
Examples & Analogies
Imagine you have 20 apples and want to divide them equally into bags that hold 7 apples each. You would fill 2 bags (making a total of 14 apples) and have 6 apples left over. Here, 20 is 'a', 7 is 'b', 2 is 'q', and 6 is 'r'.
The Fundamental Theorem of Arithmetic
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The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of primes in a unique wayβthis important fact is the Fundamental Theorem of Arithmetic.
Detailed Explanation
This theorem emphasizes that every composite number (numbers greater than one that are not prime) can be uniquely factored into prime numbers. For instance, 12 can be factored as 2 Γ 2 Γ 3. While there may be multiple ways to express it by changing the order of factors (e.g., 3 Γ 2 Γ 2), the set of prime factors remains the same. This uniqueness is critical in various areas of mathematics, including number theory.
Examples & Analogies
Think of building blocks where each block represents a prime number. You can use different arrangements to build a number like 12, but regardless of how you stack them, the specific blocks used (2, 2, and 3) determine what you've built. Just like unique addresses in a city, each composite number has its unique 'address' of prime factors.
Applications of the Theorem
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We use the Fundamental Theorem of Arithmetic for two main applications. First, we use it to prove the irrationality of many of the numbers you studied in Class IX, such as 2, 3 and 5. Second, we apply this theorem to explore when exactly the decimal expansion of a rational number is terminating and when it is non-terminating repeating.
Detailed Explanation
This chunk explains how the Fundamental Theorem of Arithmetic is not just theoretical but has practical applications. It aids in proving that certain numbers, like square roots of non-perfect squares (e.g., β2), are irrational. Additionally, it helps determine the nature of decimal expansions of rational numbers, distinguishing between terminating and repeating decimals based on the prime factors of their denominators.
Examples & Analogies
If you think of a recipe, the unique combination of ingredients (the prime factors) can yield a dish (the composite number). Just as each recipe produces a specific dish, every rational number has a particular decimal representation based on its denominator. A dish with only ingredients allowing a full measure (like whole spices) might represent a terminating decimal.
Definitions & Key Concepts
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Key Concepts
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Fundamental Theorem of Arithmetic: It establishes that every composite number has a unique prime factorization.
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Irrational Numbers: Numbers that cannot be expressed as a ratio of integers.
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HCF: The highest common factor among numbers that can help simplify fractions.
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LCM: The least common multiple used to find common grounds in combining fractions or operations.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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The number 12 can be factored as 2Β² x 3, illustrating the Fundamental Theorem of Arithmetic.
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To find the HCF of 6 and 20, we note HCF(6, 20) = 2 and LCM(6, 20) = 60.
Memory Aids
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π΅ Rhymes Time
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Fundamental math rules, keep the primes in pools.
π Fascinating Stories
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Once, a number named 12 wanted to be a prime. She discovered she had friends 2 and 3, showing her unique way to factor!
π§ Other Memory Gems
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To remember HCF and LCM: Hens Clucking Focussed - Lively Chickens Making.
π― Super Acronyms
HCF - Help Common Friends; LCM - Letβs Combine Many.
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Real Numbers
Definition:
The set of all rational and irrational numbers.
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Term: Irrational Numbers
Definition:
Numbers that cannot be expressed as a simple fraction.
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Term: Fundamental Theorem of Arithmetic
Definition:
Every composite number can be expressed uniquely as a product of prime numbers.
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Term: Composite Number
Definition:
A natural number greater than 1 that is not prime.
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Term: Prime Factorization
Definition:
The expression of a number as the product of its prime factors.
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Term: HCF (Highest Common Factor)
Definition:
The largest number that divides two or more integers without leaving a remainder.
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Term: LCM (Least Common Multiple)
Definition:
The smallest number that is a multiple of two or more integers.