Complex numbers as ordered pairs of reals - One Line Questions
1.
Let z = (2, 3). What is the ordered pair representation of i * z? —
(-3, 2)
2.
What is the conjugate of the complex number z = (3, -5) in ordered pair form? —
(3, 5)
3.
Calculate the product of z1 = (2, -1) and z2 = (-3, 5) in ordered pair form. —
(2*(-3) - (-1)*5, 2*5 + (-1)*(-3))
4.
What is the ordered pair representation of the additive inverse of z = (a, b)? —
(-a, -b)
5.
If z = (a, b), what is the ordered pair representation of i * z? —
(-b, a)
6.
What is the ordered pair representation of the complex number '-i'? —
(0, -1)
7.
What is the ordered pair representation of the complex number '-1'? —
(-1, 0)
8.
What is the multiplicative identity in the ordered pair representation of complex numbers? —
(1, 0)
9.
What is the ordered pair representation of the complex number 2 + 0i? —
(2, 0)
10.
If z = (a, b), what is the ordered pair representation of z - conjugate(z)? —
(0, 2b)
11.
What is the ordered pair representation of the complex number 'i'? —
(0, 1)
12.
What is the additive identity in the ordered pair representation of complex numbers? —
(0, 0)
13.
Let z1 = (0, 1) and z2 = (0, 1). What is z1 * z2 in ordered pair form? —
(-1, 0)
14.
Let z1 = (1, 0) and z2 = (0, 1). What is z1 * z2 in ordered pair form? —
(0, 0)
15.
Let z1 = (2, 3) and z2 = (5, 1). What is the ordered pair representation of z1 + z2? —
(7, 4)
16.
Find the multiplicative inverse of the complex number z = (2, 1) in ordered pair form. —
(2/5, -1/5)
17.
If z = (a, b), what is the ordered pair representation of z + conjugate(z)? —
(2a, 0)
18.
Let z1 = (6, 2) and z2 = (3, 5). What is the ordered pair representation of z1 - z2? —
(3, -3)
19.
Let z1 = (1, 2) and z2 = (3, 4). What is the ordered pair representation of z1 * z2? —
(1*3 - 2*4, 1*4 + 2*3)
20.
What is the ordered pair representation of the complex number 5 - 3i? —
(5, -3)
21.
The modulus of the complex number z = (3, 4) is represented as an ordered pair. What is it? —
(5, 0)
22.
The complex number 7i can be represented as an ordered pair of reals. What is this ordered pair? —
(0, 7)
23.
What is the ordered pair representation of the real number 9? —
(9, 0)
24.
The operation of complex number multiplication in ordered pair form, (a, b) * (c, d) = (ac - bd, ad + bc), is derived from expanding which expression? —
(a + ib) * (c + id)
25.
Consider the complex number z = (a, b). What is the ordered pair representation of z * conjugate(z)? —
(a^2 + b^2, 0)
26.
How is the difference of two complex numbers z1 = (a, b) and z2 = (c, d) represented as an ordered pair? —
(a-c, b-d)
27.
Consider the complex number z = (a, b). What is its conjugate represented as an ordered pair? —
(a, -b)
28.
What is the fundamental representation of a complex number in the form of an ordered pair of real numbers? —
(a, b)
29.
What is the ordered pair representation of the multiplicative inverse of a non-zero complex number z = (a, b)? —
(a/(a^2+b^2), -b/(a^2+b^2))
30.
How is the sum of two complex numbers z1 = (a, b) and z2 = (c, d) represented as an ordered pair? —
(a+c, b+d)
31.
How is the product of two complex numbers z1 = (a, b) and z2 = (c, d) represented as an ordered pair? —
(ac - bd, ad + bc)
32.
How is the complex number 'a + ib' represented as an ordered pair of reals? —
(a, b)
33.
If z = (a, b) is a complex number, then the expression a + ib is equivalent to: —
(a, b)
34.
What is the ordered pair representation of the modulus of a complex number z = (a, b)? —
(sqrt(a^2 + b^2), 0)
35.
If two complex numbers (a, b) and (c, d) are equal, what conditions must hold true? —
a = c and b = d
36.
The set of all ordered pairs of real numbers with defined addition and multiplication operations forms which algebraic structure? —
A Field
37.
Which property is satisfied by the ordered pair representation of complex numbers for addition? z1 + z2 = z2 + z1 —
Commutative Property
38.
Which property is satisfied by the ordered pair representation of complex numbers for multiplication? z1 * z2 = z2 * z1 —
Commutative Property
39.
The geometric interpretation of a complex number (a, b) is a point in which plane? —
Complex Plane
40.
Which of the following is NOT a property of complex number addition represented as ordered pairs? —
Non-distributivity
41.
Which property is satisfied by the ordered pair representation of complex numbers for addition? (z1 + z2) + z3 = z1 + (z2 + z3) —
Associative Property
42.
Which property is satisfied by the ordered pair representation of complex numbers for multiplication? (z1 * z2) * z3 = z1 * (z2 * z3) —
Associative Property
43.
Which property relates addition and multiplication for ordered pair representations of complex numbers? z1 * (z2 + z3) = z1 * z2 + z1 * z3 —
Distributive Property
44.
In the ordered pair (a, b) representing a complex number, 'a' represents which part of the complex number? —
Real part
45.
In the Argand plane, the horizontal axis represents the: —
Real part
46.
The ordered pair representation of a complex number is a direct consequence of extending the number system from real numbers to: —
Complex numbers
47.
In the ordered pair (a, b), the term 'b' is multiplied by which fundamental imaginary unit? —
i
48.
In the ordered pair (a, b) representing a complex number, 'b' represents which part of the complex number? —
Imaginary part
49.
In the Argand plane, the vertical axis represents the: —
Imaginary part
50.
Consider the complex numbers z1 = (3, 4) and z2 = (x, y). If z1 = z2, what are the values of x and y? —
x = 3, y = 4