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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. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - z?¹
Euler Formula
can't get out of the complex numbers by adding (or subtracting) or multiplying two

2. 1st. Rule of Complex Arithmetic
irrational
|z-w|
i^2 = -1
Polar Coordinates - sin?

3. V(x² + y²) = |z|
non-integers
How to solve (2i+3)/(9-i)
Polar Coordinates - r
a real number: (a + bi)(a - bi) = a² + b²

4. 3
the distance from z to the origin in the complex plane
interchangeable
i^3
can't get out of the complex numbers by adding (or subtracting) or multiplying two

5. ½(e^(-y) +e^(y)) = cosh y
i^1
cos iy
Argand diagram
Euler's Formula

6. 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.'
Complex Number
Polar Coordinates - Arg(z*)
i^4
Roots of Unity

7. x / r
Polar Coordinates - cos?
a real number: (a + bi)(a - bi) = a² + b²
integers
Real and Imaginary Parts

8. All numbers
complex
Complex Multiplication
integers
sin iy

9. The square root of -1.
Real and Imaginary Parts
Roots of Unity
Imaginary Unit
cosh²y - sinh²y

10. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
Euler Formula
v(-1)
four different numbers: i - -i - 1 - and -1.

11. y / r
Polar Coordinates - sin?
Absolute Value of a Complex Number
Irrational Number
Complex numbers are points in the plane

12. Like pi
Polar Coordinates - cos?
sin iy
Absolute Value of a Complex Number
transcendental

13. A subset within a field.
Subfield
Polar Coordinates - z
Complex Exponentiation
Complex Number Formula

14. Has exactly n roots by the fundamental theorem of algebra
the complex numbers
Field
i^4
Any polynomial O(xn) - (n > 0)

15. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Addition
Field
Square Root

16. I
cosh²y - sinh²y
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
v(-1)
Euler's Formula

17. 4th. Rule of Complex Arithmetic
ln z
Roots of Unity
(a + bi) = (c + bi) = (a + c) + ( b + d)i
(cos? +isin?)n

18. A complex number and its conjugate
i²
complex numbers
Real and Imaginary Parts
conjugate pairs

19. A number that cannot be expressed as a fraction for any integer.
The Complex Numbers
Square Root
-1
Irrational Number

20. No i
rational
zz*
real
Polar Coordinates - Multiplication by i

21. 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
sin iy
Complex Numbers: Multiply
integers

22. 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
We say that c+di and c-di are complex conjugates.
z1 / z2
-1
integers

23. 1
Imaginary number
cosh²y - sinh²y
adding complex numbers
i^2

24. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
Complex numbers are points in the plane
transcendental
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

25. I
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Field
cos iy
i^1

26. Where the curvature of the graph changes
sin z
point of inflection
Every complex number has the 'Standard Form': a + bi for some real a and b.
can't get out of the complex numbers by adding (or subtracting) or multiplying two

27. 2nd. Rule of Complex Arithmetic

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28. 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
i^0
sin z
Field

29. x + iy = r(cos? + isin?) = re^(i?)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - z
Complex Conjugate
z + z*

30. The reals are just the
Complex Exponentiation
x-axis in the complex plane
Real Numbers
Complex Number

31. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
i^1
How to add and subtract complex numbers (2-3i)-(4+6i)
Rational Number
Complex Conjugate

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

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33. 1
i^1
i^2
non-integers
z1 / z2

34. To simplify a complex fraction
cos z
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - r
point of inflection

35. E^(ln r) e^(i?) e^(2pin)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
irrational
e^(ln z)
Imaginary Unit

36. When two complex numbers are multipiled together.
integers
Polar Coordinates - z
Complex Multiplication
transcendental

37. R?¹(cos? - isin?)
z1 / z2
sin iy
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - z?¹

38. 1
Complex Numbers: Multiply
Imaginary number
|z-w|
i²

39. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
point of inflection
conjugate
Affix
Any polynomial O(xn) - (n > 0)

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

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41. When two complex numbers are divided.
z - z*
v(-1)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Division

42. 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
Imaginary Unit
integers
Real and Imaginary Parts
Complex Division

43. ? = -tan?
standard form of complex numbers
transcendental
Polar Coordinates - Arg(z*)
the distance from z to the origin in the complex plane

44. E ^ (z2 ln z1)
subtracting complex numbers
conjugate pairs
e^(ln z)
z1 ^ (z2)

45. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
0 if and only if a = b = 0
Polar Coordinates - Division
Complex Number Formula
multiplying complex numbers

46. For real a and b - a + bi =
0 if and only if a = b = 0
complex
i^2
Polar Coordinates - Multiplication

47. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Complex numbers are points in the plane
Complex Addition
Polar Coordinates - z
ln z

48. Any number not rational
Polar Coordinates - r
irrational
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - sin?

49. I^2 =
-1
interchangeable
Square Root
The Complex Numbers

50. When two complex numbers are subtracted from one another.
has a solution.
Complex Subtraction
Polar Coordinates - z
Field