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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

Instructions:

Answer 50 questions in 15 minutes.
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Match each statement with the correct term.
Don't refresh. All questions and answers are randomly picked and ordered every time you load a test.

This is a study tool. The 3 wrong answers for each question are randomly chosen from answers to other questions. So, you might find at times the answers obvious, but you will see it re-enforces your understanding as you take the test each time.

1. y / r
How to multiply complex nubers(2+i)(2i-3)
i^3
the distance from z to the origin in the complex plane
Polar Coordinates - sin?

2. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
Subfield
Complex Division
How to find any Power

3. All numbers
Polar Coordinates - z
How to add and subtract complex numbers (2-3i)-(4+6i)
zz*
complex

4. Have radical
the vector (a -b)
radicals
How to solve (2i+3)/(9-i)
Complex Number Formula

5. x / r
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Subtraction
Polar Coordinates - z
Polar Coordinates - cos?

6. (e^(-y) - e^(y)) / 2i = i sinh y
Complex Numbers: Add & subtract
i²
sin iy
a real number: (a + bi)(a - bi) = a² + b²

7. No i
real
We say that c+di and c-di are complex conjugates.
z1 ^ (z2)
Real and Imaginary Parts

8. The reals are just the
multiplying complex numbers
ln z
Rules of Complex Arithmetic
x-axis in the complex plane

9. 4th. Rule of Complex Arithmetic
cos iy
Absolute Value of a Complex Number
(a + bi) = (c + bi) = (a + c) + ( b + d)i
De Moivre's Theorem

10. V(x² + y²) = |z|
Polar Coordinates - Multiplication
Polar Coordinates - r
imaginary
Polar Coordinates - z

11. In this amazing number field every algebraic equation in z with complex coefficients
the distance from z to the origin in the complex plane
Complex Numbers: Multiply
conjugate
has a solution.

12. Equivalent to an Imaginary Unit.
Subfield
i^4
the vector (a -b)
Imaginary number

13. I^2 =
-1
The Complex Numbers
interchangeable
Euler's Formula

14. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
'i'
Argand diagram
conjugate

15. A + bi = z1 c + di = z2 - addition: z1 + z2 = (a + bi) + (c + di) = (a + c) + (b + d)i subtraction: z1 - z2 = (a - c) + (b - d)i
Complex Numbers: Add & subtract
Polar Coordinates - z
non-integers
real

16. When two complex numbers are subtracted from one another.
For real a and b - a + bi = 0 if and only if a = b = 0
The Complex Numbers
Complex Subtraction
Polar Coordinates - sin?

17. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
cos z
Complex numbers are points in the plane
For real a and b - a + bi = 0 if and only if a = b = 0
Integers

18. We can also think of the point z= a+ ib as
Polar Coordinates - z?¹
0 if and only if a = b = 0
cos z
the vector (a -b)

19. 1. i^2 = -1 2. Every complex number has the 'Standard Form': a + bi for some real a and b. 3. For real a and b - a + bi = 0 if and only if a = b = 0 4. (a + bi) = (c + bi) = (a + c) + ( b + d)i 5. (a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc
Rules of Complex Arithmetic
Field
the complex numbers
Complex Numbers: Add & subtract

20. Multiply moduli and add arguments
Subfield
Polar Coordinates - Multiplication
'i'
Polar Coordinates - z

21. E ^ (z2 ln z1)
z1 ^ (z2)
z + z*
Affix
Any polynomial O(xn) - (n > 0)

22. ½(e^(-y) +e^(y)) = cosh y
Complex Division
How to find any Power
cos iy
Polar Coordinates - z

23. xpressions such as ``the complex number z'' - and ``the point z'' are now
rational
z1 / z2
Polar Coordinates - Multiplication
interchangeable

24. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Square Root
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Numbers: Add & subtract
The Complex Numbers

25. The complex number z representing a+bi.
e^(ln z)
Imaginary Numbers
Polar Coordinates - sin?
Affix

26. Root negative - has letter i
imaginary
Complex Division
De Moivre's Theorem
Polar Coordinates - z?¹

27. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - Division
Roots of Unity

28. 5th. Rule of Complex Arithmetic
Polar Coordinates - sin?
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
standard form of complex numbers
|z| = mod(z)

29. The modulus of the complex number z= a + ib now can be interpreted as
i^2 = -1
real
the distance from z to the origin in the complex plane
Every complex number has the 'Standard Form': a + bi for some real a and b.

30. The square root of -1.
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Imaginary Unit
|z-w|
Complex Number

31. A complex number may be taken to the power of another complex number.
Complex Number
How to find any Power
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Exponentiation

32. When you subtract two complex numbers a + bi and c + di - you get the difference of the real parts and the difference of the imaginary parts: (a + bi) - (c + di) = (a - c) + (b - d)i
Polar Coordinates - sin?
subtracting complex numbers
Complex Number Formula
Complex Subtraction

33. We consider the a real number x to be the complex number x+ 0i and in this way we can think of the real numbers as a subset of
Euler Formula
the complex numbers
ln z
i²

34. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
adding complex numbers
Euler's Formula
Square Root

35. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Complex Numbers: Multiply
i^4
Roots of Unity
conjugate

36. 3rd. Rule of Complex Arithmetic
Polar Coordinates - cos?
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Number
non-integers

37. A+bi
Euler's Formula
point of inflection
Imaginary Numbers
Complex Number Formula

38. Not on the numberline
Real Numbers
Complex Conjugate
Polar Coordinates - Multiplication by i
non-integers

39. If z= a+bi is a complex number and a and b are real - we say that a is the real part of z and that b is the imaginary part of z
conjugate
the distance from z to the origin in the complex plane
Real and Imaginary Parts
The Complex Numbers

40. ? = -tan?
The Complex Numbers
(cos? +isin?)n
Polar Coordinates - Arg(z*)
z - z*

41. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
Rational Number
the vector (a -b)
sin iy

42. To simplify a complex fraction
zz*
standard form of complex numbers
the distance from z to the origin in the complex plane
multiply the numerator and the denominator by the complex conjugate of the denominator.

43. When two complex numbers are multipiled together.
z - z*
Complex Multiplication
transcendental
Liouville's Theorem -

44. I
imaginary
ln z
v(-1)
Affix

45. R?¹(cos? - isin?)
Polar Coordinates - z?¹
point of inflection
i^0
transcendental

46. Starts at 1 - does not include 0
Real and Imaginary Parts
Complex Number Formula
multiply the numerator and the denominator by the complex conjugate of the denominator.
natural

47. A + bi
zz*
standard form of complex numbers
|z| = mod(z)
z + z*

48. 1
Polar Coordinates - Arg(z*)
Complex Numbers: Multiply
i^0
point of inflection

49. A subset within a field.
natural
Polar Coordinates - Arg(z*)
Subfield
The Complex Numbers

50. 2nd. Rule of Complex Arithmetic