Test: Number Systems (Chapter 1)

Multiple Choice Questions

1. 1. The set {1, 2, 3, 4, ...} is called the set of:

   a) Whole numbers

   b) Integers

   c) Rational numbers

   d) Natural numbers

2. 2. Which number set was introduced to solve equations like x + 2 = 2?

   a) Natural numbers (N)

   b) Whole numbers (W)

   c) Integers (Z)

   d) Rational numbers (Q)

3. 3. Which number set was introduced to solve equations like x + 4 = 2?

   a) Natural numbers (N)

   b) Whole numbers (W)

   c) Integers (Z)

   d) Rational numbers (Q)

4. 4. A rational number can be expressed in the form p/q where p, q ∈ Z and:

   a) q > 0

   b) q < 0

   c) q ≠ 0

   d) q = 1

5. 5. Which of the following is an irrational number?

   a) √16

   b) 3.7

   c) √3

   d) 4/1

6. 6. A decimal which has a finite number of digits in its decimal part is called a:

   a) Recurring decimal

   b) Terminating decimal

   c) Irrational decimal

   d) Periodic decimal

7. 7. Every recurring decimal represents a(n):

   a) Irrational number

   b) Rational number

   c) Integer

   d) Whole number

8. 8. A non-terminating, non-recurring decimal represents a(n):

   a) Irrational number

   b) Rational number

   c) Integer

   d) Natural number

9. 9. The number π (Pi) is:

   a) Rational

   b) Irrational

   c) An integer

   d) Terminating decimal

10. 10. The property a + b = b + a for real numbers is called:

    a) Associative Law of Addition

    b) Additive Identity

    c) Commutative Law of Addition

    d) Additive Inverse

11. 11. The number 0 is the identity element for which operation in real numbers?

    a) Multiplication

    b) Addition

    c) Subtraction

    d) Division

12. 12. For every real number 'a', there exists '-a' such that a + (-a) = 0. '-a' is called the:

    a) Multiplicative inverse

    b) Additive identity

    c) Additive inverse

    d) Multiplicative identity

13. 13. The property a(bc) = (ab)c for real numbers is called:

    a) Commutative Law of Multiplication

    b) Associative Law of Multiplication

    c) Multiplicative Identity

    d) Distributive Law

14. 14. The number 1 is the identity element for which operation in real numbers?

    a) Multiplication

    b) Addition

    c) Subtraction

    d) Division

15. 15. For every non-zero real number 'a', there exists a⁻¹ such that a * a⁻¹ = 1. a⁻¹ is called the:

    a) Additive inverse

    b) Multiplicative inverse

    c) Multiplicative identity

    d) Additive identity

16. 16. The property a(b + c) = ab + ac is called:

    a) Commutative Law

    b) Associative Law

    c) Distributive Law (Multiplication over Addition)

    d) Closure Law

17. 17. Any set possessing the 11 field axioms (closure, associative, identity, inverse, commutative for + and *, and distributive) is called a:

    a) Group

    b) Ring

    c) Field

    d) Vector Space

18. 18. The property 'a = a' is called the ___ property of equality.

    a) Symmetric

    b) Transitive

    c) Reflexive

    d) Additive

19. 19. The property 'if a = b, then b = a' is called the ___ property of equality.

    a) Symmetric

    b) Transitive

    c) Reflexive

    d) Multiplicative

20. 20. The property 'if a = b and b = c, then a = c' is called the ___ property of equality.

    a) Symmetric

    b) Transitive

    c) Reflexive

    d) Cancellation

21. 21. Which order property states that for any two real numbers a, b, either a = b, a > b, or a < b?

    a) Transitive Property

    b) Additive Property

    c) Multiplicative Property

    d) Trichotomy Property

22. 22. If a > b and c < 0 (c is negative), then:

    a) ac > bc

    b) ac < bc

    c) ac = bc

    d) a/c > b/c

23. 23. Does the set {1, -1} have closure property with respect to addition?

    a) Yes

    b) No

    c) Only for 1

    d) Only for -1

24. 24. Does the set {1, -1} have closure property with respect to multiplication?

    a) Yes

    b) No

    c) Only for 1

    d) Only for -1

25. 25. The number 'i' (iota) is defined as:

    a) √1

    b) -1

    c) √-1

    d) 1

26. 26. A complex number is generally expressed in the form:

    a) x/y

    b) x + iy

    c) √x

    d) x!

27. 27. In the complex number z = x + iy, 'x' is called the:

    a) Imaginary part

    b) Real part

    c) Modulus

    d) Argument

28. 28. In the complex number z = x + iy, 'y' is called the:

    a) Imaginary part

    b) Real part

    c) Modulus

    d) Argument

29. 29. What is the value of i²?

    a) 1

    b) -1

    c) i

    d) -i

30. 30. What is the value of i³?

    a) 1

    b) -1

    c) i

    d) -i

31. 31. What is the value of i⁴?

    a) 1

    b) -1

    c) i

    d) -i

32. 32. The sum (a + bi) + (c + di) equals:

    a) (ac - bd) + i(ad + bc)

    b) (a + c) + i(b + d)

    c) (a - c) + i(b - d)

    d) (a + b) + i(c + d)

33. 33. The product (a + bi)(c + di) equals:

    a) (ac - bd) + i(ad + bc)

    b) (a + c) + i(b + d)

    c) (ac + bd) + i(ad - bc)

    d) (a - b) + i(c - d)

34. 34. The conjugate of the complex number z = a + bi is denoted by z̄ and equals:

    a) -a + bi

    b) -a - bi

    c) a - bi

    d) b + ai

35. 35. If z = 5 - 4i, what is its conjugate z̄?

    a) -5 - 4i

    b) -5 + 4i

    c) 5 - 4i

    d) 5 + 4i

36. 36. In the ordered pair representation, the complex number (a, b) corresponds to:

    a) a - ib

    b) b + ia

    c) a + ib

    d) b - ia

37. 37. The product of ordered pairs (a, b)(c, d) is defined as:

    a) (ac - bd, ad + bc)

    b) (ac + bd, ad - bc)

    c) (a+c, b+d)

    d) (ac, bd)

38. 38. The additive identity in the set of complex numbers C is:

    a) (1, 0)

    b) (0, 1)

    c) (0, 0)

    d) (1, 1)

39. 39. The multiplicative identity in the set of complex numbers C is:

    a) (1, 0)

    b) (0, 1)

    c) (0, 0)

    d) (1, 1)

40. 40. The complex number (0, 1) corresponds to:

    a) 1

    b) -1

    c) i

    d) -i

41. 41. The multiplicative inverse of (a, b) is:

    a) (-a, -b)

    b) (a/(a²+b²), b/(a²+b²))

    c) (a/(a²+b²), -b/(a²+b²))

    d) (-a/(a²+b²), -b/(a²+b²))

Short Answer Questions

Provide your answers in the text boxes below.

1. 1. List the progression of number systems from Natural numbers to Real numbers, explaining the need for each extension.
2. 2. Define rational numbers.
3. 3. Define irrational numbers.
4. 4. Give two examples of rational numbers and two examples of irrational numbers.
5. 5. What are terminating decimals? Do they represent rational or irrational numbers?
6. 6. What are recurring decimals? Do they represent rational or irrational numbers?
7. 7. Prove that √2 is an irrational number.
8. 8. List the five properties (laws) for addition of real numbers.
9. 9. List the five properties (laws) for multiplication of real numbers.
10. 10. State the distributive property of multiplication over addition for real numbers.
11. 11. List three properties of equality for real numbers.
12. 12. State the Trichotomy property for real numbers.
13. 13. State the Transitive property for inequalities.
14. 14. State the multiplicative properties for inequalities (consider multiplying by positive and negative numbers).
15. 15. Prove that for any real number a, a * 0 = 0.
16. 16. Prove that if ab = 0, then either a = 0 or b = 0.
17. 17. Prove that (-a)b = -ab for real numbers a, b.
18. 18. Prove that (-a)(-b) = ab for real numbers a, b.
19. 19. What is a field in mathematics?
20. 20. Define a complex number and identify its real and imaginary parts.
21. 21. What is the imaginary unit 'i' defined as?
22. 22. Calculate i⁵, i⁶, i⁷, i⁸.
23. 23. Define the conjugate of a complex number.
24. 24. If z = -2 + 3i, what is z̄?
25. 25. Define the sum of two complex numbers (a + bi) and (c + di).
26. 26. Define the product of two complex numbers (a + bi) and (c + di).
27. 27. Simplify (2 + 3i) + (4 - i).
28. 28. Simplify (1 - i) * (3 + 2i).
29. 29. How are complex numbers represented as ordered pairs?
30. 30. What is the ordered pair representation of 3 - 4i?
31. 31. What is the a + ib representation of the ordered pair (-1, 5)?
32. 32. Calculate the sum of (2, -1) and (-3, 4) as an ordered pair.
33. 33. Calculate the product of (1, 2) and (3, -1) as an ordered pair.
34. 34. What is the additive inverse of the complex number (a, b)?
35. 35. What is the multiplicative inverse of the complex number (a, b)?
36. 36. Find the multiplicative inverse of (2, -3).
37. 37. Show that the complex number (a, 0) behaves like the real number 'a' under addition and multiplication.
38. 38. Show that (0, 1) * (0, 1) = (-1, 0), justifying the definition i² = -1.