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CLEP General Mathematics: Complex Numbers

Subjects : clep, math

Instructions:

Answer 50 questions in 15 minutes.
If you are not ready to take this test, you can study here.
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. The square root of -1.
0 if and only if a = b = 0
Imaginary Unit
radicals
ln z

2. R?¹(cos? - isin?)
Polar Coordinates - Multiplication
Polar Coordinates - z?¹
a real number: (a + bi)(a - bi) = a² + b²
a + bi for some real a and b.

3. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
(cos? +isin?)n
Polar Coordinates - r
adding complex numbers

4. 3
the complex numbers
i^3
z + z*
can't get out of the complex numbers by adding (or subtracting) or multiplying two

5. 2ib
How to solve (2i+3)/(9-i)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
z - z*
multiply the numerator and the denominator by the complex conjugate of the denominator.

6. R^2 = x
Integers
point of inflection
(cos? +isin?)n
Square Root

7. E ^ (z2 ln z1)
z1 ^ (z2)
Complex Numbers: Add & subtract
interchangeable
Complex Multiplication

8. z1z2* / |z2|²
z1 / z2
irrational
|z-w|
standard form of complex numbers

9. 1
How to solve (2i+3)/(9-i)
Real Numbers
i^4
sin iy

10. Like pi
transcendental
z + z*
i^1
standard form of complex numbers

11. Real and imaginary numbers
i^1
'i'
complex numbers
integers

12. zn = (cos? + isin?)n = cosn? + isinn? - For all integers n

13. 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
Every complex number has the 'Standard Form': a + bi for some real a and b.
i^3
non-integers

14. To simplify a complex fraction
Integers
How to find any Power
ln z
multiply the numerator and the denominator by the complex conjugate of the denominator.

15. Where the curvature of the graph changes
imaginary
the complex numbers
point of inflection
Absolute Value of a Complex Number

16. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
the vector (a -b)
a real number: (a + bi)(a - bi) = a² + b²
Rational Number
How to add and subtract complex numbers (2-3i)-(4+6i)

17. In the same way that we think of real numbers as being points on a line - it is natural to identify a complex number z=a+ib with the point (a -b) in the cartesian plane.
Complex numbers are points in the plane
i^1
Square Root
Polar Coordinates - cos?

18. The complex number z representing a+bi.
a + bi for some real a and b.
Affix
-1
0 if and only if a = b = 0

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
radicals
i²
Argand diagram
Rules of Complex Arithmetic

20. A + bi
standard form of complex numbers
cos z
cosh²y - sinh²y
Polar Coordinates - Multiplication by i

21. E^(i?) = cos? + isin? ; e^(ip) + 1 = 0

22. Multiply moduli and add arguments
cosh²y - sinh²y
Polar Coordinates - Multiplication
Polar Coordinates - Arg(z*)
radicals

23. In this amazing number field every algebraic equation in z with complex coefficients
z + z*
has a solution.
conjugate pairs
How to find any Power

24. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Division
Complex Number Formula
x-axis in the complex plane

25. (a + bi) = (c + bi) =
the complex numbers
i^2
Every complex number has the 'Standard Form': a + bi for some real a and b.
(a + c) + ( b + d)i

26. 2nd. Rule of Complex Arithmetic

27. I
integers
Imaginary number
|z| = mod(z)
i^1

28. y / r
Polar Coordinates - sin?
Field
Euler's Formula
Complex Conjugate

29. A subset within a field.
Polar Coordinates - Multiplication
conjugate pairs
Subfield
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

30. 1
Complex Addition
complex
i^0
Polar Coordinates - z?¹

31. When two complex numbers are divided.
complex numbers
Complex Division
zz*
Argand diagram

32. ? = -tan?
|z-w|
De Moivre's Theorem
Polar Coordinates - Arg(z*)
Complex Exponentiation

33. ½(e^(iz) + e^(-iz))
Polar Coordinates - z
(a + c) + ( b + d)i
cos z
radicals

34. Numbers on a numberline
integers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
|z| = mod(z)
Complex Exponentiation

35. Not on the numberline
e^(ln z)
conjugate pairs
non-integers
Real Numbers

36. Every complex number has the 'Standard Form':
complex numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
Imaginary Unit
a + bi for some real a and b.

37. (2+i)(2i-3) you would use the foil methom which is first outter inner last. (2x2i)(2x-3)(ix2i^2)(ix(-3) =i-8
For real a and b - a + bi = 0 if and only if a = b = 0
How to multiply complex nubers(2+i)(2i-3)
Imaginary Unit
Complex Number

38. 1
Rational Number
i²
Imaginary Numbers
For real a and b - a + bi = 0 if and only if a = b = 0

39. 1
Complex Multiplication
i^2
imaginary
Imaginary Unit

40. V(x² + y²) = |z|
Polar Coordinates - sin?
Polar Coordinates - r
For real a and b - a + bi = 0 if and only if a = b = 0
How to solve (2i+3)/(9-i)

41. 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
Real and Imaginary Parts
complex numbers
Complex Exponentiation
Every complex number has the 'Standard Form': a + bi for some real a and b.

42. When two complex numbers are added together.
Field
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Addition
Any polynomial O(xn) - (n > 0)

43. To prove that number field every algebraic equation in z with complex coefficients has a solution we need

44. Given (4-2i) the complex conjugate would be (4+2i)
i^2 = -1
imaginary
Square Root
Complex Conjugate

45. No i
real
|z| = mod(z)
Polar Coordinates - Arg(z*)
Real Numbers

46. When two complex numbers are subtracted from one another.
multiplying complex numbers
Complex Subtraction
Any polynomial O(xn) - (n > 0)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

47. We can also think of the point z= a+ ib as
the vector (a -b)
Roots of Unity
The Complex Numbers
z - z*

48. All the powers of i can be written as
Complex Addition
transcendental
cos z
four different numbers: i - -i - 1 - and -1.

49. I
i^0
v(-1)
(cos? +isin?)n
Euler Formula

50. Divide moduli and subtract arguments
Polar Coordinates - Division
i^2
interchangeable
Complex Conjugate