important points of the Number Systems chapter

Here are the important points of the Number Systems chapter for 11th Class Mathematics (FBISE syllabus):

1. Types of Numbers:
   • Natural Numbers: Positive integers (1, 2, 3, ...).
   • Whole Numbers: Natural numbers including zero (0, 1, 2, 3, ...).
   • Integers: Positive, negative numbers, and zero (..., -2, -1, 0, 1, 2, ...).
   • Rational Numbers: Numbers expressible as a fraction

     \frac{p}{q}

     , where ( p ) and ( q ) are integers and

     q \neq 0

     (e.g.,

     \frac{1}{2}, 0.75

     ).
   • Irrational Numbers: Numbers that cannot be expressed as a fraction, e.g.,

     \sqrt{2}, \pi

     .
   • Real Numbers: All numbers on the number line, including rational and irrational numbers.
   • Complex Numbers: Numbers of the form

     a + bi

     , where ( a ) and ( b ) are real numbers, and

     i = \sqrt{-1}

     .
2. Properties of Real Numbers:
   • Closure: The sum, difference, product, and quotient (except division by zero) of real numbers are real numbers.
   • Commutative:

     a + b = b + a

     ,

     a \cdot b = b \cdot a

     .
   • Associative:

     (a + b) + c = a + (b + c)

     ,

     (a \cdot b) \cdot c = a \cdot (b \cdot c)

     .
   • Distributive:

     a(b + c) = ab + ac

     .
   • Identity: Additive identity is 0 (

     a + 0 = a

     ); multiplicative identity is 1 (

     a \cdot 1 = a

     ).
   • Inverse: Additive inverse is

     -a

     (

     a + (-a) = 0

     ); multiplicative inverse is

     \frac{1}{a}

     (

     a \cdot \frac{1}{a} = 1

     ,

     a \neq 0

     ).
3. Complex Numbers:
   • Standard Form:

     z = a + bi

     , where ( a ) is the real part, and ( b ) is the imaginary part.
   • Conjugate: The conjugate of

     z = a + bi

     is

     \overline{z} = a - bi

     .
   • Operations: Addition, subtraction, multiplication, and division of complex numbers.
   • Modulus:

     |z| = \sqrt{a^2 + b^2}

     .
   • Argument: The angle

     \theta = \tan^{-1}\left(\frac{b}{a}\right)

     in polar form.
   • Polar Form:

     z = r(\cos\theta + i\sin\theta)

     , where

     r = |z|

     .
4. Number Line and Intervals:
   • Real numbers can be represented on a number line.
   • Intervals: Open (( (a, b) )), closed (( [a, b] )), and half-open (( [a, b) ) or ( (a, b] )).
5. Radicals and Exponents:
   • Radicals:

     \sqrt[n]{a} = a^{1/n}

     , where ( n ) is the index, and ( a ) is the radicand.
   • Laws of Exponents:
     • a^m \cdot a^n = a^{m+n}
     • \frac{a^m}{a^n} = a^{m-n}
     • (a^m)^n = a^{mn}
     • a^{-n} = \frac{1}{a^n}
   • Simplification of radical expressions.
6. Inequalities and Absolute Value:
   • Absolute Value:

     |x| = x

     if

     x \geq 0

     , and

     |x| = -x

     if

     x < 0

     .
   • Solving inequalities involving absolute values, e.g.,

     |x - a| < b

     .
7. Key Concepts for FBISE:
   • Representation of numbers in different forms (decimal, fraction, etc.).
   • Conversion between rational and irrational forms.
   • Solving problems involving complex number operations.
   • Understanding properties and their applications in proofs.

Tips for FBISE Exams:

• Focus on solving exercises from the NBF textbook, especially problems involving complex number operations and properties of real numbers.
• Practice MCQs related to number system classifications and properties, as they are common in FBISE exams.
• Use resources like MathCity.org or ClassNotes.xyz for solved examples and additional practice.

If you need detailed notes or specific examples for any topic, let me know!

------------------------------------------------------------

Below is a set of Multiple Choice Questions (MCQs) for Chapter 1: Number Systems (11th Class Mathematics, FBISE syllabus) along with their solutions. These MCQs cover key concepts such as types of numbers, properties of real numbers, complex numbers, and radicals, tailored to the FBISE curriculum.

MCQs for Chapter 1: Number Systems

1. Which of the following is an irrational number?
   a)

   \frac{2}{3}

   
   b)

   \sqrt{16}

   
   c)

   \sqrt{2}

   
   d) 0.75
   Answer: c)

   \sqrt{2}

   
   Solution: An irrational number cannot be expressed as a fraction

   \frac{p}{q}

   (where ( p ) and ( q ) are integers,

   q \neq 0

   ).
   • \frac{2}{3}

     is rational (fraction form).
   • \sqrt{16} = 4

     , which is rational.
   • \sqrt{2} \approx 1.414

     is irrational as it cannot be written as a fraction.
   • 0.75 is rational (

     \frac{3}{4}

     ).
2. The conjugate of the complex number

   3 - 4i

   is:
   a)

   -3 + 4i

   
   b)

   3 + 4i

   
   c)

   -3 - 4i

   
   d)

   4 - 3i
   Answer: b)

   3 + 4i

   
   Solution: The conjugate of a complex number

   a + bi

   is

   a - bi

   . For

   3 - 4i

   , the conjugate is

   3 + 4i

   .
3. Which property of real numbers is illustrated by

   (2 + 3) + 5 = 2 + (3 + 5)

   ?
   a) Commutative property
   b) Associative property
   c) Distributive property
   d) Identity property
   Answer: b) Associative property
   Solution: The associative property states that the grouping of numbers in addition does not affect the result:

   (a + b) + c = a + (b + c)

   . Here,

   (2 + 3) + 5 = 2 + (3 + 5)

   .
4. The modulus of the complex number

   2 + 3i

   is:
   a)

   \sqrt{13}

   
   b)

   \sqrt{5}

   
   c) 5
   d) 13
   Answer: a)

   \sqrt{13}

   
   Solution: The modulus of a complex number

   a + bi

   is

   |z| = \sqrt{a^2 + b^2}

   . For

   2 + 3i

   ,

   |z| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}

   .
5. Which of the following is a whole number but not a natural number?
   a) 1
   b) 0
   c) -1
   d) 2
   Answer: b) 0
   Solution: Whole numbers include 0 and all natural numbers ({0, 1, 2, 3, \ldots}). Natural numbers are {1, 2, 3, \ldots}. Thus, 0 is a whole number but not a natural number.
6. Simplify:

   \sqrt{50} \div \sqrt{2}

   
   a) 5
   b)

   \sqrt{25}

   
   c)

   \sqrt{48}

   
   d) 25
   Answer: a) 5
   Solution: Using the property of radicals,

   \frac{\sqrt{50}}{\sqrt{2}} = \sqrt{\frac{50}{2}} = \sqrt{25} = 5

   .
7. The value of

   i^4

   is:
   a) 1
   b) -1
   c) ( i )
   d)

   -i
   Answer: a) 1
   Solution: The imaginary unit

   i = \sqrt{-1}

   , and its powers follow a cycle:
   • i^1 = i

     ,

     i^2 = -1

     ,

     i^3 = -i

     ,

     i^4 = 1

     .
     Thus,

     i^4 = 1

     .
8. Which of the following is a rational number?
   a)

   \pi

   
   b)

   \sqrt{3}

   
   c)

   \frac{5}{2}

   
   d)

   \sqrt{7}
   Answer: c)

   \frac{5}{2}

   
   Solution: A rational number can be expressed as

   \frac{p}{q}

   , where ( p ) and ( q ) are integers and

   q \neq 0

   .
   • \pi

     and

     \sqrt{3}, \sqrt{7}

     are irrational.
   • \frac{5}{2}

     is rational.
9. The additive inverse of

   -7

   is:
   a) 7
   b) -7
   c) 0
   d)

   \frac{1}{7}
   Answer: a) 7
   Solution: The additive inverse of a number ( a ) is

   -a

   , such that

   a + (-a) = 0

   . For

   -7

   , the additive inverse is ( 7 ).
10. The product of

    (2 + 3i)(2 - 3i)

    is:
    a) 13
    b)

    4 + 9i

    
    c)

    4 - 9i

    
    d) 7
    Answer: a) 13
    Solution: For complex numbers,

    (a + bi)(a - bi) = a^2 + b^2

    .
    Here,

    (2 + 3i)(2 - 3i) = 2^2 + 3^2 = 4 + 9 = 13

    .

Additional Resources

For more practice, you can access MCQs with solutions from the following websites aligned with the FBISE syllabus:

• MathCity.org: Offers solved MCQs from past FBISE papers for Chapter 1.
• ClassNotes.xyz: Provides chapter-wise MCQs with answers and explanations.
• FG Study: Contains a large database of MCQs for Number Systems, designed for FBISE students.
• IlmkiDunya.com: Offers online MCQ tests for Chapter 1 with answers.

Tips for Preparation:

• Practice these MCQs to familiarize yourself with question patterns in FBISE exams.
• Review the explanations to understand mistakes and reinforce concepts.

• Solve past papers available on MathCity.org or FG Study for additional practice.