Mathematics - Chapter - 1 - Number System Prime Number Notes

Number System

Main Concept and their results:

• Large Numbers up to One Crore:
• Main Concept: Understanding and working with numbers in the range of 1 to 10,000,000 (one crore).
• Result: Proficiency in reading, writing, and performing operations with large numbers.
• Example: 9,874,562 is read as "Nine million, eight hundred seventy-four thousand, five hundred sixty-two."
• Reading and Writing of Large Numbers:
• Main Concept: The ability to correctly read and write large numbers in words and digits.
• Result: Improved communication of numerical values.
• Example: 6,543,210 is written as "Six million, five hundred forty-three thousand, two hundred ten."
• Comparing Large Numbers:
• Main Concept: Determining the relationships (greater than, less than, equal to) between large numbers.
• Result: Enhanced number sense and mathematical reasoning.
• Example: 7,890,123 > 6,543,210 (read as "Seven million, eight hundred ninety thousand, one hundred twenty-three is greater than six million, five hundred forty-three thousand, two hundred ten.")
• Indian System of Numeration:
• Main Concept: Understanding and using the traditional Indian numbering system (place value system).
• Result: Proficiency in representing numbers in the Indian system.
• Example: In the Indian system, 34,567 is written as ३४,५६७.
• International System of Numeration:
• Main Concept: Understanding and using the international numbering system (decimal system).
• Result: Proficiency in representing numbers in the international system.
• Example: In the international system 34,567 is written as 34,567.
• Use of Large Numbers:
• Main Concept: Applying large numbers in real-world scenarios, such as in finance, science, and statistics.
• Result: Practical problem-solving skills.
• Example: Calculating the national debt of a country, this can run into trillions of units in the international system.
• Estimation of Numbers:
• Main Concept: Approximating the value of numbers to make quick calculations.
• Result: Efficient mental math and estimation skills.
• Example: Estimating that 38,729 is approximately 40,000 for a rough calculation.
• Use of Brackets:
• Main Concept: Understanding and correctly using parentheses () and brackets [] in mathematical expressions.
• Result: Clearer and accurate mathematical notation.
• Example: (5 + 3) x 2 = 16 and [6 - (4 + 2)] = 0.
• Roman Numerals:
• Main Concept: Understanding and working with Roman numerals.
• Result: Familiarity with historical numerical notation.
• Example: IX represents 9, and XLVIII represents 48 in Roman numerals.
• Lemma
• A lemma is a proven statement used for proving another statement.
• Euclid's Division Lemma
• If given positive integers $a$ and $b$, there exist unique integers $q$ and $r$ satisfying $a = bq + r$, where $0\leq r<b$.

Here we call ‘$a$’ as dividend, ‘$b$’ as divisor, ‘$q$’ as quotient and ‘$r$’ as remainder.

$\therefore \text {Dividend} = \left(\text{Divisor} \times \text {Quotient}\right) + \text {Remainder}$

If in Euclid’s lemma $r = 0$ then $b$ would be HCF of  ‘$a$’ and ‘$b$’.

• Integers
• All natural numbers, negative of natural numbers and 0, together are called integers. i.e. ………. – 3, – 2, – 1, 0, 1, 2, 3, 4, ………….. are integers.
• Algorithm
• An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.
• Euclid's Division Algorithm
• Euclid’s division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers. Recall that the HCF of two positive integers $a$ and $b$ is the largest positive integer $d$ that divides both $a$ and $b$.

Whole Numbers:

• Definition: Whole numbers are a set of numbers that include all natural numbers (positive counting numbers) along with zero.
• Example: {0, 1, 2, 3, 4, ...}
• Natural Numbers:
• Definition: Natural numbers are positive counting numbers starting from 1.
• Example: {1, 2, 3, 4, ...}
• Predecessor and Successor of a Natural Number:
• Predecessor: The number that comes just before a given natural number.
• Example: The predecessor of 7 is 6.
• Successor: The number that comes just after a given natural number.
• Example: The successor of 7 is 8.
• Representation of Whole Numbers on the Number Line:
• Explanation: Whole numbers can be represented as points on a number line, with each number's position corresponding to its value.
• Example: On a number line, 0, 1, 2, 3, and 4 would be marked at equally spaced intervals.
• Addition and Subtraction of Whole Numbers on the Number Line:
• Addition: Move to the right on the number line to add.
• Example: To add 3 and 2, start at 3 and move two units to the right to reach 5.
• Subtraction: Move to the left on the number line to subtract.
• Example: To subtract 2 from 5, start at 5 and move two units to the left to reach 3.
• Properties of Whole Numbers:
• Closure Property: The sum or product of two whole numbers is also a whole number.
• Commutativity: Changing the order of addition or multiplication doesn't change the result.
• Example: 3 + 5 = 5 + 3 and 3 × 5 = 5 × 3.
• Associativity: The way numbers are grouped in addition or multiplication doesn't change the result.
• Example: (2 + 3) + 4 = 2 + (3 + 4) and (2 × 3) × 4 = 2 × (3 × 4).
• Distributivity: Multiplication distributes over addition.
• Example: 2 × (3 + 4) = (2 × 3) + (2 × 4).
• Identities: The additive identity is 0 (a + 0 = a), and the multiplicative identity is 1 (a × 1 = a).
• Division of a Whole Number by Zero is Not Defined:
• Explanation: Division by zero is undefined because it leads to mathematical inconsistency.
• Example: 5 ÷ 0 is not a valid mathematical operation.
• Patterns in Whole Numbers:
• Explanation: Whole numbers often exhibit various patterns, such as arithmetic progressions (e.g., 2, 4, 6, 8...) and geometric progressions (e.g., 2, 4, 8, 16...).
• Example: The sequence 3, 7, 11, 15... forms an arithmetic progression with a common difference of 4.

Some Important Points related to Number

• Factors and Multiples:
• Factors: These are numbers that can divide another number without leaving a remainder.
• Multiples: These are numbers that are obtained by multiplying a number by an integer.
• Example : Factors of 12 are 1, 2, 3, 4, 6, and 12. Multiples of 5 are 5, 10, 15, 20, ...
• Perfect Number:
• Definition: A perfect number is one for which the sum of all its factors (excluding itself) is equal to twice the number.
• Example : 28 is a perfect number because its factors (1, 2, 4, 7, 14) sum up to 28, which is twice the number.
• Prime and Composite Numbers:
• Prime Number: A number greater than 1 that has only two factors, 1 and itself.
• Composite Number: A number greater than 1 that has more than two factors.
• Example : 7 is a prime number because it has only two factors, 1 and 7. 8 is a composite number because it has factors 1, 2, 4, and 8.
• Tests for Divisibility:
• These are rules to determine if one number is divisible by another.
• Example : To check if 356 is divisible by 4, you can use the rule for divisibility by 4: If the last two digits (56) form a number divisible by 4, then the entire number is divisible by 4. Since 56 is divisible by 4, 356 is also divisible by 4.
• Common Factors and Common Multiples:
• Common Factors: Factors that two or more numbers have in common.
• Common Multiples: Multiples that two or more numbers share.
• Example : Common factors of 12 and 18 are 1, 2, 3, and 6. Common multiples of 4 and 6 are 12, 24, 36, ...
• Coprime Numbers:
• Definition: Numbers that have no common factors other than 1.
• Example : 15 and 28 are coprime because their only common factor is 1.
• More Divisibility Rules:
• These rules provide shortcuts for determining divisibility.
• Example : To check if a number is divisible by 9, add up its digits. If the sum is divisible by 9, the original number is also divisible by 9.
• Prime Factorization of a Number:
• Expressing a number as the product of its prime factors.
• Example : Prime factorization of 24: 24 = 2 × 2 × 2 × 3.
• Highest Common Factor (HCF) and Least Common Multiple (LCM):
• HCF: The largest common factor of two or more numbers.
• LCM: The smallest multiple that is a common multiple of two or more numbers.
• Example : HCF of 18 and 24 is 6. LCM of 18 and 24 is 72.
• Use of HCF and LCM in Real-Life Problems:
• HCF and LCM are used in various practical situations, like finding the least number of items to be bought for a certain number of people.
• Example : If you need to distribute 18 chocolates to students and 24 chocolates to another group, you can find the least number of chocolates you need to buy to ensure each group gets an equal number by finding the LCM of 18 and 24, which is 72. So, you need to buy 72 chocolates.

Fundamental Theorem of Arithmetic:

"Every composite number (a positive integer greater than 1 that is not prime) can be expressed (factorized) as a unique product of prime numbers. This unique factorization holds true regardless of the order in which the prime factors are listed."

Explanation:

• Every Composite Number:
• Composite numbers are positive integers greater than 1 that have more than two distinct positive divisors. In other words, they can be divided evenly by numbers other than 1 and themselves.
• For example- 4, 6, 8, and 9 are composite numbers.
• Expressed as a Product of Primes:
• The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of prime numbers.
• For example- Take the composite number 12.
• Factorizing 12 into its prime factors: 12 = 2 × 2 × 3
• Here, we have expressed 12 as a product of prime numbers: 2 and 3.
• Unique Factorization:
• The theorem further asserts that this factorization is unique. In other words, there is only one way to write a composite number as a product of prime factors.
• For example - no matter how we rearrange the factors of 12, we will always end up with the same prime factors. For instance:
• 12 = 2 × 2 × 3
• 12 = 3 × 2 × 2
• 12 = 2 × 3 × 2  ...and so on.
• In each case, the prime factors remain the same: 2 and 3.
• It ensures that prime numbers are the building blocks for all composite numbers and that each composite number has a unique prime factorization.

Application of Fundamental Theorem of Arithmetic:

• It states that every composite number can be uniquely expressed as a product of prime numbers.
• It has numerous applications in various areas of mathematics and science.
• Some applications are:
• Highest Common Factor (HCF) and Least Common Multiple (LCM):
• The prime factorization method is often used to find the HCF and LCM of two or more numbers.
• Example: Find the GCD and LCM of 24 and 36.
• Prime factorization of 24: 2^3 * 3^1
• Prime factorization of 36: 2^2 * 3^2
• HCF: Take the minimum exponent of each prime factor: HCF (24, 36) = 2² x 3¹ = 12
• LCM: Take the maximum exponent of each prime factor: LCM (24, 36) = 2³ x 3² = 72
• Simplifying Fractions:
• Prime factorization helps in simplifying fractions by canceling common factors.
• Example: Simplify 18/24.
• Prime factorization of 18: 2¹ x 3² = 2 x 3 x 3
• Prime factorization of 24: 2³ x 3¹ = 2 x 2 x 2 x 3
• Cancel common factors: 
  
• Checking for Prime Numbers:
• We can use the Fundamental Theorem to check whether a number is prime or composite by examining its prime factorization.
• Example: Is 17 a prime or composite number?
• If a number is prime, its only prime factorization is itself.
• In the case of 17, it has no other prime factors except 17, so it's a prime number.
• Divisibility Tests:
• The unique prime factorization allows us to derive divisibility tests for various numbers.
• Example: To check if a number is divisible by 9, sum its digits. If the sum is divisible by 9, the number itself is divisible by 9.
• Example: To check if a number is divisible by 4, examine its last two digits. If they form a number divisible by 4, the original number is divisible by 4.
• Counting Divisors:
• The number of divisors of a composite number can be determined by examining its prime factorization.
• Example: How many divisors does 60 have?
• Prime factorization of 60: 2² x 3¹ x 5¹
• To find the number of divisors, add 1 to each exponent and multiply: (2+1) x (1+1) x (1+1) = 3 x 2 x 2 = 12 divisors.
• Number Systems:
• Prime factorization is fundamental in understanding different number systems, such as rational numbers, decimals, and complex numbers.
• Example: Expressing recurring decimals as fractions relies on prime factorization to find patterns and simplify.

Prime Factorization Method:

• Step 1: Start with the given composite number.
• Step 2: Begin by dividing the number by the smallest prime number, which is 2, and continue dividing until you can no longer divide evenly.
• Step 3: Move on to the next prime number (3), and again, divide the result from the previous step until you can no longer divide evenly.
• Step 4: Continue this process, moving to the next prime number each time, until you have factored the composite number completely.
• Step 5: Write down the prime factors in ascending order.

Example: Find the prime factorization of the composite number 36 using the prime factorization method.

Solution :

• Step 1: Start with the number 36.
• Step 2: Divide 36 by the smallest prime number, which is 2:  36 ÷ 2 = 18 So, we have found that 2 is a prime factor, and we continue with the result, which is 18.
• Step 3: Divide 18 by 2 again: 18 ÷ 2 = 9      2 is a prime factor, and we continue with 9.
• Step 4: Divide 9 by the next prime number, which is 3: 9 ÷ 3 = 3     3 is a prime factor, and we continue with 3.
• Step 5: Divide 3 by 3: 3 ÷ 3 = 1       3 is a prime factor, and we have now completely factored the number.
• Thus, we write down the prime factors in ascending order:
• Prime Factorization of 36 = 2 × 2 × 3 × 3

So, the prime factorization of 36 is 2² × 3², where the exponents 2 and 2 indicate that the prime factor 2 is raised to the power of 2, and the prime factor 3 is raised to the power of 2. This is the unique representation of 36 as a product of its prime factors.

The relationship between the HCF and the LCM for two positive integers:

The Fundamental Theorem of Arithmetic says that for any two positive integers a and b, their HCF multiplied by their LCM will always be equal to the product of the two integers, which is a × b. This property is useful in various mathematical applications, including simplifying fractions and solving equations involving factors and multiples.

Example:

• Consider two positive integers, a = 12 and b = 18.
• HCF (a, b):
• To find the HCF of 12 and 18, we need to list their common factors and identify the largest one.
• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 18: 1, 2, 3, 6, 9, 18
• The common factors are 1, 2, 3, and 6. The largest common factor is 6.
• So, HCF(12, 18) = 6.
• LCM (a, b):
• To find the LCM of 12 and 18, we need to list their multiples and identify the smallest one that both numbers share.
• Multiples of 12: 12, 24, 36, 48, 60, ...
• Multiples of 18: 18, 36, 54, 72, ...
• The smallest multiple that both numbers share is 36.
• So, LCM(12, 18) = 36.
• Now, let's use the Fundamental Theorem of Arithmetic to verify if it holds true:
• HCF (a, b) × LCM (a, b) = a × b
• HCF(12, 18) × LCM(12, 18) = 6 × 36 = 216
• a × b = 12 × 18 = 216
• As we can see, HCF(12, 18) × LCM(12, 18) equals a × b, which confirms the validity of Theorem.

The relationship between the HCF and the LCM for Three positive integers:

• L.C.M. (a, b, c) x HCF (a, b, c) = a x b x c

Here a, b, c, are positive integers.

The product of the LCM and the HCF of three numbers is equal to the product of the three numbers themselves.

• For Expression : HCF (a, b, c) x LCM (a, b, c) ¹ a x b x c

Why above expressions are not equal because:

1. HCF(a, b, c): The Highest Common Factor (HCF) is the largest positive integer that divides all three numbers a, b and c without leaving a remainder.
2. LCM(a, b, c): The Least Common Multiple (LCM) is the smallest positive integer that is a multiple of all three numbers a, b and c.

Consider the expression - HCF (a, b, c) x LCM (a, b, c)

This expression represents the product of the HCF and the LCM of the numbers a, b and c.

If a,b and c are coprime (i.e., they have no common factors other than 1), then the HCF of a, b and c is 1. In this case, HCF (a, b, c) =1  and the expression becomes:

1 x LCM (a, b, c) = LCM (a, b, c)

So, when the numbers a, b and c are coprime, the left side of the expression simplifies to the LCM of a,b  and c.

However, when a, b and c have common factors greater than 1, the HCF will be greater than 1, and the expression HCF (a, b, c) x LCM (a, b, c) will be larger than the product  a x b x c.

In short, the two expressions are not equal a, b and c have common factors greater than 1. This is because the HCF accounts for these common factors, while the product   a x b x c  does not.

Irrational Numbers:

An irrational number is a real number that cannot be written as a fraction of two integers, where the denominator is not zero, and its decimal expansion is non-terminating and non-repeating.

Examples of Irrational Numbers:

• π (Pi):
• Pi (π) is one of the most famous irrational numbers. It represents the ratio of the circumference of a circle to its diameter.
• Its decimal representation begins with 3.14159265359... and goes on forever without repeating.
• It cannot be expressed as a simple fraction.
• √2 (Square Root of 2):
• The square root of 2 (√2) is another common irrational number.
• Its decimal representation starts with 1.41421356237... and continues infinitely without any repeating pattern.
• √2 cannot be expressed as a fraction of two integers.
• e (Euler's Number):
• Euler's number (e) is an essential irrational number in mathematics, frequently encountered in calculus and exponential growth.
• Its decimal representation starts with 2.71828182845... and extends infinitely without repeating.
• Like other irrational numbers, e cannot be expressed as a fraction of integers.
• √3 (Square Root of 3):
• The square root of 3 (√3) is yet another example of an irrational number.
• Its decimal expansion starts with 1.73205080757... and goes on infinitely without repetition.
• √3 cannot be simplified into a fraction of integers.
• φ (Phi - Golden Ratio):
• The golden ratio (φ) is an irrational number often encountered in art, architecture, and nature.
• Its decimal representation starts with 1.61803398875... and continues indefinitely without repeating.
• It cannot be expressed as a simple fraction.
• √5 (Square Root of 5):
• The square root of 5 (√5) is an irrational number.
• Its decimal representation starts with 2.2360679775... and goes on infinitely without repeating.
• √7 (Square Root of 7):
• The square root of 7 (√7) is another irrational number.
• Its decimal expansion begins with 2.6457513110... and continues infinitely without repeating.
• √10 (Square Root of 10):
• The square root of 10 (√10) is also irrational.
• Its decimal representation starts with 3.1622776601... and extends indefinitely without repetition.
• √13 (Square Root of 13):
• The square root of 13 (√13) is yet another example of an irrational number.
• Its decimal expansion begins with 3.6055512755... and continues infinitely without repeating.
• √17 (Square Root of 17):
• The square root of 17 (√17) is an irrational number.
• Its decimal representation starts with 4.1231056256... and goes on forever without a repeating pattern.
• √19 (Square Root of 19):
• The square root of 19 (√19) is another example of an irrational number.
• Its decimal expansion begins with 4.3588989435... and extends infinitely without repetition.
• √29 (Square Root of 29):
• The square root of 29 (√29) is an irrational number.
• Its decimal representation starts with 5.3851648071... and continues infinitely without repeating.
• √31 (Square Root of 31):
• The square root of 31 (√31) is yet another irrational number.
• Its decimal expansion begins with 5.5677643628... and goes on forever without a repeating pattern.

Theorem 1  : 

Let p be a prime number. If p divides a², then p divides a, where a is a positive integer.

Proof :

• Prime Factorization of a: 

a = p₁ p₂ ... p_n, where p₁, p₂, ..., p_n are prime numbers, not necessarily distinct.

• Expression of a²:

Next, we calculate a² by squaring a: 

a² = (p₁ p₂  ... p_n)(p₁ p₂  ... p_n) = p₁²  p₂²  ...  p_n².

• Given p Divides a²:

Since the theorem states that p divides a², it means p is one of the prime factors of a².

• Application of Fundamental Theorem of Arithmetic:

We invoke the Fundamental Theorem of Arithmetic, which states that every positive integer has a unique prime factorization.

• Uniqueness of Prime Factors of a²:

We recognize that the only prime factors of a² are p₁, p₂, ..., p_n, as per its prime factorization.

• Conclusion -

p is Among p₁, p₂, ..., p_n:

We conclude that p must be one of the prime factors among p₁, p₂, ..., p_n.

• Implication for a:

Finally, since a = p₁  p₂  ...  p_n, it implies that p divides a.

• This concludes the proof of Theorem, which demonstrates that if a prime number p divides the square of a positive integer a, then p must also divide a.
•  The proof method used here is based on "proof by contradiction," which is a technique that assumes the opposite of what we want to prove and then shows that it leads to a contradiction. This technique helps establish the truth of the original statement.

Theorem 2 :    √2 is irrational.

Proof:

Assume that √2 is a rational number. This means that √2 can be expressed as a fraction of two integers in the simplest form as √2 = a/b where a and b are integers with no common factors other than 1.

• Square Both Sides:
  

Now, square both sides of the equation to isolate √2:

(√2)² = (a/b)²

2 = (a²/b²)

• Rearrange:

Rearrange the equation to solve for a²:

a² = 2b²       where a² is an even number (since it's divisible by 2).

• Odd or Even:

Consider whether "a" is even or odd:

If "a" is even, we can write it as a = 2k, where k is an integer.

By Substituting it in the equation a² = 2b² yields (2k)² = 2b², which simplifies to 4k² = 2b².

• Divide both sides by 2: 2k² = b².

This implies that "b" is also even since it's equal to 2 times another integer.

If "a" is odd, we can write it as a = 2k + 1, where k is an integer.

Substituting this into the equation a² = 2b² yields (2k + 1)² = 2b².

Expand and simplify: 4k² + 4k + 1 = 2b².

Rearrange: 2(2k² + 2k) + 1 = 2b².

Notice that the left side is odd (2 times an integer plus 1), which means that "b²" must also be odd.

• Contradiction:

We have reached a contradiction. In both cases, we have found that "b²" is even when it should be odd (since "a² = 2b²" implies "b²" is even). This contradiction arises from our initial assumption that √2 can be expressed as a fraction of two coprime integers.

• Thus we can say:

Since assuming √2 is rational leads to a contradiction, we conclude that √2 is irrational. This means that √2 cannot be expressed as a simple fraction of two integers, and its decimal representation goes on infinitely without repeating.

√3 is irrational

Assume that √3 is a rational number. This means that √3 can be expressed as a fraction of two integers in the simplest form: √3 = a/b   where a and b are integers with no common factors other than 1

• Square Both Sides: 

Now, square both sides of the equation to isolate √3:

(√3)² = (a/b)²

3 = (a²/b²)

• Rearrange:

Rearrange the equation to solve for a²:

a² = 3b²

Now, we have a² = 3b², where a² is divisible by 3 (since it's equal to 3 times another integer).

• Divisibility by 3:

Consider whether "a" is divisible by 3 or not:

If "a" is not divisible by 3, it means that "a" can be written as a = 3k + 1 or a = 3k + 2, where k is an integer.

• Substituting these into the equation a² = 3b² yields:

For a = 3k + 1: (3k + 1)² = 3b²

• Expand and simplify: 9k² + 6k + 1 = 3b²
• Rearrange: 3(3k² + 2k) + 1 = 3b²

Notice that the left side is not divisible by 3 (3 times an integer plus 1), which means that "b²" must also not be divisible by 3.

For a = 3k + 2, a similar argument applies.

If "a" is divisible by 3, it means a = 3m, where m is an integer.

Substituting this into the equation a² = 3b² yields: (3m)² = 3b²

• Simplify: 9m² = 3b²
• Divide both sides by 3: 3m² = b²

This implies that "b" is also divisible by 3 since it's equal to 3 times another integer.

• Contradiction:

In all cases, we have reached a contradiction. In both scenarios, we have found that "b²" is divisible by 3 when it should not be (since "a² = 3b²" implies "b²" is not divisible by 3). This contradiction arises from our initial assumption that √3 can be expressed as a fraction of two coprime integers.

• Conclusion:

Since assuming √3 is rational leads to a contradiction, we conclude that √3 is irrational. This means that √3 cannot be expressed as a simple fraction of two integers, and its decimal representation goes on infinitely without repeating.

5 - √3 is irrational

Assume, for the sake of contradiction, that 5 - √3 is rational. This means that 5 - √3 can be expressed as a fraction of two integers in the simplest form:

where  a and b are integers with no common factors other than 1 (i.e., they are coprime).

• Isolate √3

Now, isolate √3 on one side of the equation:

• Square Both Sides:

Next, square both sides of the equation:

• Rearrange:

Rearrange the equation

• Both Sides are Rational:

Notice that both sides of this equation are rational numbers because they are expressed as fractions of integers.

Conclusion:

Since assuming 5 - √3 is rational leads to a contradiction, we conclude that 5 - √3 is irrational. This means that 5 - √3 cannot be expressed as a simple fraction of two integers, and its decimal representation goes on infinitely without repeating.

3 √2 is irrational

Assume, for the sake of contradiction, that 3 √2 is rational. This means it can be expressed as a ratio of two integers a and b in its simplest form (i.e., a fraction where a and b have no common factors other than 1)

3 √2 = a/b

Square both sides of the equation:

(3√2)² = a²/b²

18 = a²/b²

• Rearrange the equation to isolate a²:

a² = 18 . b²

We observe that the right side of the equation is a multiple of 18, which means that a² must also be a multiple of 18.

This implies that a must be a multiple of 3 because a² being a multiple of 18 implies that a itself is a multiple of √18 which simplifies to 3 √2.

So, a can be expressed as a=3k for some integer k.

• Substituting this back into the equation:

(3k)² = 18 . b²

9k² = 18 . b²

• Divide both sides by 9:

K² = 2 . b²

We see that k² is even, which means k must also be even because the square of an odd integer is odd.

Let k = 2m for some integer m.

• Substituting this back into the equation:

(2m)² = 2 . b²

4m² = 2 . b²

• Divide both sides by 2:

2m² = b²

Now, we see that b² is also even, which means b must also be even.

However, if both a and b are even, they have a common factor of 2, which contradicts our initial assumption that a and b have no common factors other than 1 (i.e., they are coprime). This contradiction arises from assuming that 3 √2 is rational.

Conclusion:

Therefore, we conclude that 3 √2 is irrational because it cannot be expressed as a simple fraction of two integers, and it cannot be written in the form a/b  where a and b are coprime integers.