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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. All numbers
complex
Complex Multiplication
Complex Division
i^0

2. All the powers of i can be written as
rational
Polar Coordinates - sin?
How to multiply complex nubers(2+i)(2i-3)
four different numbers: i - -i - 1 - and -1.

3. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Polar Coordinates - Arg(z*)
the vector (a -b)
Complex Numbers: Add & subtract
Field

4. (a + bi) = (c + bi) =
Square Root
(a + c) + ( b + d)i
x-axis in the complex plane
sin iy

5. 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.
0 if and only if a = b = 0
Complex Numbers: Add & subtract
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex numbers are points in the plane

6. 4th. Rule of Complex Arithmetic
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - Division
(a + bi) = (c + bi) = (a + c) + ( b + d)i
real

7. Where the curvature of the graph changes
Field
point of inflection
adding complex numbers
interchangeable

8. 3
Complex Number Formula
Imaginary number
z - z*
i^3

9. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
sin z
-1
(a + c) + ( b + d)i

10. V(zz*) = v(a² + b²)
Polar Coordinates - Multiplication
conjugate
Affix
|z| = mod(z)

11. (a + bi)(c + bi) =
Any polynomial O(xn) - (n > 0)
zz*
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(a + c) + ( b + d)i

12. A plot of complex numbers as points.
Rules of Complex Arithmetic
Argand diagram
i^2 = -1
irrational

13. ½(e^(iz) + e^(-iz))
Complex Addition
transcendental
cos z
radicals

14. The field of numbers of the form - where and are real numbers and i is the imaginary unit equal to the square root of - . When a single letter is used to denote a complex number - it is sometimes called an 'affix.'
(a + c) + ( b + d)i
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Number
Every complex number has the 'Standard Form': a + bi for some real a and b.

15. Divide moduli and subtract arguments
-1
zz*
0 if and only if a = b = 0
Polar Coordinates - Division

16. V(x² + y²) = |z|
Imaginary Numbers
Polar Coordinates - r
How to add and subtract complex numbers (2-3i)-(4+6i)
Affix

17. The reals are just the
x-axis in the complex plane
Complex Addition
Complex Subtraction
Field

18. Derives z = a+bi
Rational Number
the distance from z to the origin in the complex plane
Complex numbers are points in the plane
Euler Formula

19. E^(ln r) e^(i?) e^(2pin)
Complex Multiplication
e^(ln z)
Square Root
cos z

20. A number that can be expressed as a fraction p/q where q is not equal to 0.
complex
the complex numbers
Any polynomial O(xn) - (n > 0)
Rational Number

21. 1
(cos? +isin?)n
cosh²y - sinh²y
x-axis in the complex plane
Real Numbers

22. The square root of -1.
Polar Coordinates - sin?
Imaginary Numbers
Imaginary Unit
i^1

23. 1
Real and Imaginary Parts
i^0
De Moivre's Theorem
Euler Formula

24. Root negative - has letter i
i^1
Polar Coordinates - Multiplication by i
imaginary
(a + bi) = (c + bi) = (a + c) + ( b + d)i

25. 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
z1 / z2
z + z*
Complex Multiplication

26. R^2 = x
Square Root
sin iy
(a + bi) = (c + bi) = (a + c) + ( b + d)i
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

27. (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
imaginary
Imaginary number
How to solve (2i+3)/(9-i)
Every complex number has the 'Standard Form': a + bi for some real a and b.

28. No i
real
transcendental
ln z
How to find any Power

29. When two complex numbers are subtracted from one another.
Complex Subtraction
Square Root
integers
Real and Imaginary Parts

30. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - sin?
Euler Formula
i^4

31. In this amazing number field every algebraic equation in z with complex coefficients
Euler Formula
i^1
Integers
has a solution.

32. I = imaginary unit - i² = -1 or i = v-1
How to add and subtract complex numbers (2-3i)-(4+6i)
Imaginary Numbers
Rational Number
Polar Coordinates - z?¹

33. (e^(-y) - e^(y)) / 2i = i sinh y
Real Numbers
irrational
natural
sin iy

34. For real a and b - a + bi =
Imaginary Unit
Imaginary Numbers
0 if and only if a = b = 0
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

35. ? = -tan?
Polar Coordinates - Arg(z*)
Polar Coordinates - cos?
i^1
Field

36. 2nd. Rule of Complex Arithmetic

37. Notice that rules 4 and 5 state that we complex numbers together - we can divide by c + di if c and d are not both zero. But there is a much easier way to do division.

38. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
e^(ln z)
ln z
Absolute Value of a Complex Number
Euler's Formula

39. R?¹(cos? - isin?)
Real Numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - z?¹
Complex Division

40. 1st. Rule of Complex Arithmetic
can't get out of the complex numbers by adding (or subtracting) or multiplying two
imaginary
Polar Coordinates - Multiplication by i
i^2 = -1

41. Real and imaginary numbers
multiplying complex numbers
complex numbers
conjugate
subtracting complex numbers

42. A subset within a field.
subtracting complex numbers
Subfield
cos z
four different numbers: i - -i - 1 - and -1.

43. Imaginary number

44. A number that cannot be expressed as a fraction for any integer.
How to solve (2i+3)/(9-i)
integers
The Complex Numbers
Irrational Number

45. Like pi
a real number: (a + bi)(a - bi) = a² + b²
transcendental
Polar Coordinates - Arg(z*)
z + z*

46. Starts at 1 - does not include 0
four different numbers: i - -i - 1 - and -1.
radicals
natural
conjugate

47. I^2 =
How to multiply complex nubers(2+i)(2i-3)
natural
-1
i^3

48. 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: Multiply
Roots of Unity
four different numbers: i - -i - 1 - and -1.
Complex Numbers: Add & subtract

49. 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
radicals
imaginary
|z-w|
Real and Imaginary Parts

50. What about dividing one complex number by another? Is the result another complex number? Let's ask the question in another way. If you are given four real numbers a -b -c and d - can you find two other real numbers x and y so that
Square Root
How to add and subtract complex numbers (2-3i)-(4+6i)
We say that c+di and c-di are complex conjugates.
a real number: (a + bi)(a - bi) = a² + b²