Real Numbers — Class 10 Maths Notes

Topics covered in Real Numbers

1. 1.What Are Real Numbers?

   Real Numbers: Real numbers include all the numbers that we use in everyday life. This category includes all the numbers you can think of, like:

   1. Whole numbers (like 0, 1, 2,...)
   2. Integers (which includes negative numbers like -1, -2,...)
   3. Fractions (like ½, ¾,...)
   4. Decimal numbers (like 3.14, 0.5,...)
   5. Irrational numbers (like the square root of 2, π (pi), which cannot be written as a simple fraction or decimal)

   Real Life Example: Think about your age, 14 years. That's a real number. If you measure the length of your pencil and it's 16.5 cm, that's also a real number.

   Use in Career: Real numbers are used everywhere. If you become an engineer, you'll use them to calculate dimensions and forces. In finance, you'll use them for calculating interest and profits. Basically, any field that involves measuring, counting, or calculating will use real numbers. What is not Real Numbers?

   A number that is not a real number falls into a different category called "imaginary" or "complex" numbers. Let me explain:

   Imaginary Numbers

   • Imaginary numbers are numbers that can't be found on the number line that we use for real numbers.
   • The most basic imaginary number is the square root of -1, which is represented as i. In real numbers, you can't have the square root of a negative number, so this is where imaginary numbers come in.
   • For example, −4−4​ is not a real number. Instead, it's written as 22i, using the imaginary unit i.

   Complex Numbers

   • Complex numbers are a combination of real and imaginary numbers.
   • They are usually written in the form +a+bi, where a is a real number, and bi is an imaginary number.
   • For example, 3+23+2i is a complex number.

   Real Life Context

   In real life, especially in higher-level mathematics, physics, and engineering, complex numbers are very useful. They are used to describe things like electrical currents and waves that can't be represented with just real numbers. The Fundamental Theorem of Arithmetic

   
   Think of prime numbers as the building blocks of all numbers. Prime numbers are numbers greater than 1 that can only be divided evenly by themselves and 1 (e.g., 2, 3, 5, 7). The Fundamental Theorem of Arithmetic tells us that every whole number greater than 1 can be built using only these building blocks, and there's only one way to build each number.
   
   

   Easy Explanation:

   Imagine you have a toy box filled with different kinds of blocks, but some of them are special and can't be split into smaller pieces. These special blocks are like our prime numbers.

   Now, let's say you want to build something with a number like 12. You can only make it by using exactly three 2's and one 3. You can't make 12 with any other combination of these special blocks.

   Numerical Example:

   Let's look at the number 30: 30 = 3 x 10 Now, we can break down 10 into its prime factors: 10 = 2 x 5 So, 30 can be expressed as: 30 = 3 x 2 x 5
   
   

   You can't make 30 with any other combination of prime numbers!

   Real-life Example:

   Think of a cake recipe that requires exactly 3 eggs, 2 cups of sugar, and 5 tablespoons of butter. There's only one way to make this specific cake with these exact amounts. This is kind of like how there's only one way to make the number 30 with its specific prime numbers.
   Where It Will Be Used in Real Life and Career:
   

   
   

   Playing Games: Some puzzles and games involve prime numbers, and understanding this concept can make you a better player.

   
   

   Computer Stuff: If you ever want to be a computer whiz, understanding prime numbers can help you with computer security and making computer programs.

   
   

   Being a Math Detective: Just like detectives solve mysteries, mathematicians can figure out the "mystery" of a number by breaking it down into prime numbers.

2. 2.Find HCF & LCM in a easy way of any numbers

   Finding HCF (Highest Common Factor)
   

   Steps to Find HCF by Prime Factorization Method:

   1. List Prime Factors: Write down the prime factors of each number.

   2. Common Factors: Identify the common prime factors between the numbers.

   3. Multiply: Multiply the common prime factors.

   Example:

   Find HCF of 12 and 18.

   1. Prime factors of 12: 2 x 2 x 3

   2. Prime factors of 18: 2 x 3 x 3

   3. Common factors: 2, 3

   HCF = 2 x 3 = 6

   Real-Life Example:

   You have 12 chocolates and your friend has 18. You both want to make equal groups. The largest group you can make will have 6 chocolates, which is the HCF.

   ---

   Finding LCM (Least Common Multiple)

   Steps to Find LCM by Prime Factorization Method:

   1. List Prime Factors: Write down the prime factors of each number.

   2. Highest Power: Take the highest power of all prime factors.

   3. Multiply: Multiply those highest power factors.

   Example:

   Find LCM of 12 and 18.

   1. Prime factors of 12: 2 x 2 x 3

   2. Prime factors of 18: 2 x 3 x 3

   3. Highest power factors: 2², 3²

   LCM = 4 x 9 = 36

   Real-Life Example:

   You take 12 minutes to finish a game and your friend takes 18 minutes. You both can play together again after 36 minutes, which is the LCM.

3. 3.Find the LCM and HCF of 6 and 20 by the prime factorisation method

   Step 1: Prime Factorization

   First, we'll break down each number into its prime factors.

   - 12: 2 × 2 × 3

   - 40: 2 × 2 × 2 × 5

   Step 2: Finding HCF

   For the HCF, take the common prime factors and multiply them.

   - Common prime factors of 12 and 40: 2 × 2 = 4

   So, the HCF is 4.
   
   

   Step 3: Finding LCM
   
   

   For the LCM, take all the prime factors from both numbers (don't repeat the common ones) and multiply them.

   
   

   - All prime factors: 2 × 2 × 3 × 2 × 5 = 120

   So, the LCM is 120.

   Real-Life Example

   Imagine you have 12 chocolates and your friend has 40 chocolates. You both want to divide your chocolates into equal groups without breaking any chocolate. The largest number of chocolates that can be in each group without breaking any is the HCF, which is 4 in this case.

   And if you both want to combine your chocolates and then divide them into equal groups, the smallest number of chocolates that can be in each group is the LCM, which is 120 in this case.

4. 4.Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.

   Step 1: Prime Factorization

   First, we'll find the prime factors of each number.

   - 16: 2 × 2 × 2 × 2

   - 72: 2 × 2 × 2 × 3 × 3

   - 160: 2 × 2 × 2 × 2 × 5 × 5

   Step 2: Finding HCF

   For HCF, we take the common prime factors from all three numbers and multiply them.

   - Common prime factors of 16, 72, and 160: 2 × 2 × 2 = 8

   So, the HCF is 8.

   Step 3: Finding LCM

   For LCM, we take all the prime factors from all three numbers (don't repeat the common ones) and multiply them.

   - All prime factors: 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5 = 7200

   So, the LCM is 7200.

   Real-Life Example

   Imagine you, your friend, and a neighbor have 16, 72, and 160 marbles respectively. You all want to make equal-sized groups of marbles without splitting any marble. The largest number of marbles that can be in each group is the HCF, which is 8 in this case.

   If you all combine your marbles and then want to divide them into equal groups, the smallest number of marbles that can be in each group is the LCM, which is 7200 in this case.

5. 5.Consider the numbers 8n , where n is a natural number. Check whether there is any value of n for which 8n ends with the digit zero.

   We're asked to find out if there is a natural number in such that 8n ends with the digit zero.
   
   

   For a number to end in zero, it has to be a multiple of 10. This means that 8n should be divisible by 10
   
   

   To check if 8m can be a multiple of 10, let's break down 10 into its prime factors: 10= 2 X 5.

   
   8n = 8 X n = 23
   
   

   For 8n to be divisible by 10, it should have both 2 and 5 as factors
   
   

   We already have 23 (or three factors of 2) in 8n, but we don't have a 5.
   
   

   Therefore, in must be at least 5 for 8n to end with the digit zero. So yes, there is a value of n for which 8 ends with the digit zero, and that value is - 5.

6. 6.Expressing Numbers as a Product of Prime Factors

   Prime Factors

   1. 180

   - 180 = 2 x 90

   - 90 = 2 x 45

   - 45 = 5 x 9

   - 9 = 3 x 3

   So, 180 = 2 x 2 x 3 x 3 x 5

   2. 256

   - 256 = 2 x 128

   - 128 = 2 x 64

   - 64 = 2 x 32

   - 32 = 2 x 16

   - 16 = 2 x 8

   - 8 = 2 x 4

   - 4 = 2 x 2

   So, 256 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 (or 2⁸)

   3. 5005

   - 5005 = 5 x 1001

   - 1001 = 7 x 143

   - 143 = 11 x 13

   So, 5005 = 5 x 7 x 11 x 13

7. 7.Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers.

   1. 25 and 71
   
   

   SOLUTION
   
   

   Find LCM and HCF

   
   

   HCF: 1 (They are co-prime numbers)
   
   

   LCM: 1775 (25 x 71)
   
   

   Verification: LCM x HCF = 1775 x 1 = 1775
   
   

   Product of numbers: 25 x 71 = 1775
   
   

   So, LCM x HCF = Product of numbers
   
   2. 610 and 92
   
   SOLUTION
   
   

   HCF: 2
   
   

   LCM: 28130 (610 x 92 / 2)
   
   

   Verification: LCM x HCF = 28130 x 2 = 56260
   
   

   Product of numbers: 610 x 92 = 56260
   
   

   So, LCM x HCF = Product of numbers

8. 8.Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

   Explanation for Composite Numbers

   1. For the expression 7 x 11 x 13 + 13:
   
   - Calculation: 7 x 11 x 13 = 1001
   

   - Adding 13: 1001 + 13 = 1014

   
   

   1014 can be divided by 2, so it has more than two factors (1, 2, 507, 1014). Therefore, 1014 is a composite number.

   
   
   

   2. For the expression 7 x 6 x 5 x 4 x 3 x 2 x 1 + 5:
   
   

   - Calculation: 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040

   - Adding 5: 5040 + 5 = 5045
   
   

   5045 can be divided by 5, so it has more than two factors (1, 5, 1009, 5045). Therefore, 5045 is a composite number.

9. 9.Revisiting Irrational Numbers

   What are Irrational Numbers?

   Irrational numbers are numbers that can't be written as a simple fraction. Examples are square root of 2, Pi, and Euler's Number (e).

   How to Identify?

   1. Square Roots: If square root of a number is not a whole number, it's irrational. Like square root of 2.

   2. Pi: The value of Pi is around 3.14159 and it never ends or repeats. So it's irrational.

   3. Non-Repeating Decimals: Any decimal that never ends or repeats is irrational.

   Why called Irrational?

   They are called irrational because they can't be written as a ratio of two integers.

   Real-Life Example

   Imagine measuring the diagonal of a square with sides of 1 unit. The length would be square root of 2 units. You can't measure it exactly because square root of 2 is irrational.

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

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

    What Does the Statement Mean?

    1. p is a prime number, like 2, 3, 5, 7, 11, and so on.
    
    

    2. a is a positive integer, like 1, 2, 3, 4, and so on.
    
    

    3. If p can divide a² (meaning a² divided by p leaves no remainder), then p must also be able to divide a
    
    

    Simple Explanation

    Let's say you have a number a and you square it to get a² Now, if a prime number p can perfectly divide a², then it should also perfectly divide a.

    Example

    Let's take ( p = 3) (a prime number) and ( a = 6) (a positive integer).

    1. a² = 6 x 6 = 36
    
    2. P divides a² because 36 divided by 3 leaves no remainder.

    3. According to the statement, p should also divide a. And yes, 6 divided by 3 also leaves no remainder.

    So, the statement is true!

    Why Is This Important?

    This concept is crucial in number theory and helps us understand the properties of numbers, especially when dealing with prime numbers.
    

11. 11.Prove that √3 is irrational

    1. Assumption: You assumed √3 is rational, meaning it can be expressed as a fraction a/b, where a and b are integers with no common factors (other than 1), and b is not zero.

    2. Equation: You stated √3 = a/b.

    3. Squaring Both Sides: You correctly squared both sides to get 3 = a² / b².

    4. Multiply by b²: This step led to 3b² = a².

    5. Observation: You noted that since a² is 3 times b², it means a², and therefore a, is divisible by 3.

    6. Let a = 3c: Here, you introduced a new integer c, and rightly expressed a as 3c.

    7. Substitute a: Substituting a = 3c into the equation gave 3b² = (3c)², which simplifies to 3b² = 9c².

    8. Divide by 3: This step led to b² = 3c², indicating b², and therefore b, is also divisible by 3.

    9. Contradiction: You identified a contradiction here. The initial assumption was that a and b have no common factors other than 1, but both are divisible by 3.

    10. Conclusion: This contradiction implies the initial assumption that √3 is rational is incorrect. Therefore, √3 is irrational.

12. 12.Prove that 3+2√5 is irrational.

    Step 1: Assumption

    Assume that 3 + 2√5 is rational. This means it can be written as a/b, where a and b are integers and b is not zero.

    Step 2: Equation

    So, 3 + 2√5 = a/b

    Step 3: Multiply by b

    b(3 + 2√5) = a

    Step 4: Expand

    3b + 2b√5 = a

    Step 5: Isolate √5

    2b√5 = a - 3b

    Step 6: Divide by 2b

    √5 = (a - 3b) / 2b

    Step 7: Observation

    Here, √5 is written as (a - 3b) / 2b, which means √5 would be rational. But we know that √5 is irrational.

    Step 8: Conclusion

    Our initial assumption that 3 + 2√5 is rational is incorrect. Therefore, 3 + 2√5 is irrational.

13. 13.Prove that 3√2 is irrational.

    1. Assumption: Let's start by assuming that 3232​ is rational. This means that it can be written as a fraction ba​, where a and b are whole numbers and b is not zero.

    2. Equation: This gives us the equation 32=32​=ba​.

    3. Clearing the Fraction: To get rid of the fraction, we multiply both sides by b: 32×= 32​×b=
       a

    4. Simplifying: This simplifies to 32=3b2​=a.

    5. Square Both Sides: Squaring both sides of the equation gives (32)2=2(3b2​)2=a2, which leads to 92×2=29b2×2=a2 or 182=218b2=a2.

    6. Contradiction: Notice that the left side of the equation 18218b2 is an even number, which makes 2a2 also an even number. Therefore, a must be even. If a is even, then =2a=2n for some whole number n.

    7. Substitution: Substituting =2a=2n into 182=218b2=a2 gives 182=4218b2=4n2 or 92=229b2=2n2.

    8. Contradiction Again: The right side 222n2 is an even number, making 929b2 also even. Therefore, b must also be even.

    9. Conclusion: Both a and b are even, which contradicts our initial assumption that ba​ is in its simplest form (because both cannot be even in the simplest form). Therefore, our original assumption that 3232​ is rational must be wrong. Hence, 3232​ is irrational.

14. 14.Quick Revision

    1. Real Numbers: Real numbers include all the numbers we see and use in daily life. They consist of both rational numbers (like 2, ½, 0.75) and irrational numbers (like √2, π). In simple terms, if you can put it on the number line, it's a real number.

    2. HCF & LCM:

    • HCF (Highest Common Factor): It's the largest number that can divide two or more numbers without leaving any remainder. For example, the HCF of 12 and 16 is 4.
    • LCM (Least Common Multiple): It's the smallest number that is a multiple of two or more numbers. For example, the LCM of 12 and 16 is 48.

    3. Revisiting Irrational Numbers: Irrational numbers are real numbers that cannot be written as a simple fraction. They have non-repeating and non-ending decimal parts. For example, π (3.14159...) and √2 (approximately 1.41421...) are irrational numbers.

    Important Formulas and Implementation

    1. Finding HCF:

    • Formula: There isn't a direct formula for HCF, but it's often found using the Euclidean algorithm or prime factorization.
    • Example: To find the HCF of 18 and 24, list their prime factors: 18 = 2 × 3 × 3, 24 = 2 × 2 × 2 × 3. The common factors are 2 and 3, so HCF = 2 × 3 = 6.

    2. Finding LCM:

    • Formula: LCM of two numbers = (Product of the numbers) / HCF of those numbers.
    • Example: For 18 and 24, LCM = (18 × 24) / HCF (6) = 432 / 6 = 72.