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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. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
Complex Exponentiation
Polar Coordinates - cos?
Polar Coordinates - z?¹

2. All numbers
sin z
Imaginary Numbers
complex
De Moivre's Theorem

3. When you add two complex numbers a + bi and c + di - you get the sum of the real parts and the sum of the imaginary parts: (a + bi) + (c + di) = (a + c) + (b + d)i
adding complex numbers
Complex Number
integers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

4. Every complex number has the 'Standard Form':
i^1
conjugate pairs
a + bi for some real a and b.
the vector (a -b)

5. Not on the numberline
complex numbers
cosh²y - sinh²y
non-integers
v(-1)

6. A number that can be expressed as a fraction p/q where q is not equal to 0.
Subfield
Complex numbers are points in the plane
Rational Number
Complex Number Formula

7. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
multiply the numerator and the denominator by the complex conjugate of the denominator.
Imaginary number
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

8. 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
e^(ln z)
four different numbers: i - -i - 1 - and -1.
subtracting complex numbers
i²

9. x + iy = r(cos? + isin?) = re^(i?)
Euler Formula
Polar Coordinates - z
sin iy
i²

10. Given (4-2i) the complex conjugate would be (4+2i)
How to multiply complex nubers(2+i)(2i-3)
has a solution.
Complex Conjugate
0 if and only if a = b = 0

11. Root negative - has letter i
Complex Addition
imaginary
(cos? +isin?)n
Affix

12. ½(e^(-y) +e^(y)) = cosh y
Roots of Unity
cos iy
Rules of Complex Arithmetic
0 if and only if a = b = 0

13. Like pi
Affix
conjugate
transcendental
Complex Multiplication

14. xpressions such as ``the complex number z'' - and ``the point z'' are now
Complex Number
interchangeable
adding complex numbers
cos z

15. 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
Polar Coordinates - Multiplication
De Moivre's Theorem
Polar Coordinates - r

16. A complex number may be taken to the power of another complex number.
Complex Exponentiation
How to find any Power
i^2
cos z

17. Numbers on a numberline
Complex Number
interchangeable
How to add and subtract complex numbers (2-3i)-(4+6i)
integers

18. In this amazing number field every algebraic equation in z with complex coefficients
complex
has a solution.
non-integers
ln z

19. 1
De Moivre's Theorem
i^0
Complex Subtraction
Complex Exponentiation

20. A plot of complex numbers as points.
integers
Every complex number has the 'Standard Form': a + bi for some real a and b.
i^0
Argand diagram

21. I
cos iy
v(-1)
imaginary
Complex Numbers: Multiply

22. ? = -tan?
Liouville's Theorem -
(cos? +isin?)n
Polar Coordinates - Arg(z*)
Real and Imaginary Parts

23. The reals are just the
The Complex Numbers
conjugate
x-axis in the complex plane
Square Root

24. The field of all rational and irrational numbers.
Integers
adding complex numbers
Real Numbers
How to multiply complex nubers(2+i)(2i-3)

25. R^2 = x
multiplying complex numbers
Square Root
Affix
z1 ^ (z2)

26. The modulus of the complex number z= a + ib now can be interpreted as
the distance from z to the origin in the complex plane
e^(ln z)
standard form of complex numbers
z - z*

27. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
complex
multiplying complex numbers
i^2 = -1
x-axis in the complex plane

28. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
i²
Polar Coordinates - sin?
Argand diagram
Integers

29. 1
i^3
i^4
conjugate
i^2 = -1

30. When two complex numbers are added together.
|z-w|
Complex Addition
De Moivre's Theorem
i^2

31. A number that cannot be expressed as a fraction for any integer.
i^3
the complex numbers
Irrational Number
Polar Coordinates - z

32. When two complex numbers are divided.
sin z
Complex Division
0 if and only if a = b = 0
Real and Imaginary Parts

33. (2i+3)/(9-i)for the denominator you multiply by the conjugate and what u do to the bottom u have to do to the top then you distribute the bottom then the top then add like terms then you simplify. 21i+25/17
Complex Multiplication
Imaginary Numbers
Polar Coordinates - Arg(z*)
How to solve (2i+3)/(9-i)

34. 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
|z| = mod(z)
Complex Number Formula
complex numbers

35. (e^(iz) - e^(-iz)) / 2i
i²
sin iy
sin z
For real a and b - a + bi = 0 if and only if a = b = 0

36. 2ib
conjugate pairs
z - z*
How to multiply complex nubers(2+i)(2i-3)
Euler's Formula

37. The complex number z representing a+bi.
radicals
x-axis in the complex plane
(a + c) + ( b + d)i
Affix

38. (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
How to multiply complex nubers(2+i)(2i-3)
Irrational Number
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - Multiplication by i

39. No i
Imaginary Unit
real
integers
Polar Coordinates - z

40. Equivalent to an Imaginary Unit.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
four different numbers: i - -i - 1 - and -1.
Complex Numbers: Add & subtract
Imaginary number

41. 2a
a + bi for some real a and b.
z + z*
Polar Coordinates - cos?
Complex Number

42. Where the curvature of the graph changes
|z| = mod(z)
Real and Imaginary Parts
point of inflection
e^(ln z)

43. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
radicals
Liouville's Theorem -
Roots of Unity
irrational

44. Formula: z1 · z2 = (a + bi)(c + di) = ac +adi +cbi +bdi² = (ac - bd) + (ad +cb)i - when you multiply a complex number by its conjugate - you get a real number.
Complex Subtraction
Complex Numbers: Multiply
Rational Number
Complex numbers are points in the plane

45. A + bi
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - Multiplication by i
Rational Number
standard form of complex numbers

46. 3
Rules of Complex Arithmetic
'i'
i^3
Square Root

47. For real a and b - a + bi =
Complex Number
i^1
0 if and only if a = b = 0
Imaginary Unit

48. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Absolute Value of a Complex Number
Complex Exponentiation
-1
transcendental

49. y / r
zz*
Complex Numbers: Multiply
Polar Coordinates - sin?
four different numbers: i - -i - 1 - and -1.

50. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
integers
Complex Subtraction
i^4
The Complex Numbers