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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. (e^(-y) - e^(y)) / 2i = i sinh y
Polar Coordinates - Multiplication
Real and Imaginary Parts
sin iy
radicals

2. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
rational
Field
Complex Number Formula
Complex Exponentiation

3. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
irrational
Roots of Unity
cosh²y - sinh²y
radicals

4. All the powers of i can be written as
multiply the numerator and the denominator by the complex conjugate of the denominator.
sin z
Polar Coordinates - Division
four different numbers: i - -i - 1 - and -1.

5. Any number not rational
zz*
irrational
Polar Coordinates - sin?
How to solve (2i+3)/(9-i)

6. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
cos z
Polar Coordinates - z
four different numbers: i - -i - 1 - and -1.

7. All numbers
z1 ^ (z2)
Imaginary Unit
complex
Absolute Value of a Complex Number

8. I^2 =
-1
'i'
Polar Coordinates - z
cos z

9. 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
Every complex number has the 'Standard Form': a + bi for some real a and b.
complex
Complex Numbers: Add & subtract
z + z*

10. 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.
the vector (a -b)
subtracting complex numbers
Complex Numbers: Multiply
real

11. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Real Numbers
i^1
How to add and subtract complex numbers (2-3i)-(4+6i)
Complex Division

12. Has exactly n roots by the fundamental theorem of algebra
Complex Division
Complex Number Formula
adding complex numbers
Any polynomial O(xn) - (n > 0)

13. When two complex numbers are divided.
x-axis in the complex plane
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
i^2
Complex Division

14. Divide moduli and subtract arguments
Polar Coordinates - Division
four different numbers: i - -i - 1 - and -1.
complex
Square Root

15. Like pi
Integers
How to add and subtract complex numbers (2-3i)-(4+6i)
sin iy
transcendental

16. We can also think of the point z= a+ ib as
Any polynomial O(xn) - (n > 0)
the vector (a -b)
multiplying complex numbers
|z-w|

17. A complex number and its conjugate
Subfield
Every complex number has the 'Standard Form': a + bi for some real a and b.
Rational Number
conjugate pairs

18. A number that cannot be expressed as a fraction for any integer.
Integers
radicals
Irrational Number
cos z

19. 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
i^2 = -1
subtracting complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
-1

20. Starts at 1 - does not include 0
Imaginary number
i^1
natural
rational

21. R^2 = x
Complex Number Formula
0 if and only if a = b = 0
complex numbers
Square Root

22. V(x² + y²) = |z|
a real number: (a + bi)(a - bi) = a² + b²
sin iy
Polar Coordinates - r
v(-1)

23. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiplying complex numbers
sin z
Complex Numbers: Multiply
can't get out of the complex numbers by adding (or subtracting) or multiplying two

24. Every complex number has the 'Standard Form':
How to find any Power
a + bi for some real a and b.
Complex Exponentiation
standard form of complex numbers

25. 1st. Rule of Complex Arithmetic
interchangeable
i^2 = -1
a + bi for some real a and b.
Polar Coordinates - Division

26. y / r
i^4
For real a and b - a + bi = 0 if and only if a = b = 0
De Moivre's Theorem
Polar Coordinates - sin?

27. 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
Polar Coordinates - Arg(z*)
Complex Addition
Real Numbers

28. A² + b² - real and non negative
cosh²y - sinh²y
Real and Imaginary Parts
zz*
Rules of Complex Arithmetic

29. 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
the complex numbers
i^1
Complex Multiplication
Complex Numbers: Add & subtract

30. The complex number z representing a+bi.
Complex Numbers: Multiply
multiplying complex numbers
Affix
the complex numbers

31. To simplify the square root of a negative number
cosh²y - sinh²y
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - Multiplication by i
0 if and only if a = b = 0

32. A subset within a field.
Complex Division
Polar Coordinates - Multiplication by i
Euler's Formula
Subfield

33. The modulus of the complex number z= a + ib now can be interpreted as
cos iy
zz*
Complex Division
the distance from z to the origin in the complex plane

34. Imaginary number

35. Equivalent to an Imaginary Unit.
rational
We say that c+di and c-di are complex conjugates.
the vector (a -b)
Imaginary number

36. 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
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Number
sin z
We say that c+di and c-di are complex conjugates.

37. z1z2* / |z2|²
z1 / z2
Complex Addition
adding complex numbers
De Moivre's Theorem

38. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
a real number: (a + bi)(a - bi) = a² + b²
(a + c) + ( b + d)i
Absolute Value of a Complex Number
Any polynomial O(xn) - (n > 0)

39. x + iy = r(cos? + isin?) = re^(i?)
For real a and b - a + bi = 0 if and only if a = b = 0
cos z
Polar Coordinates - z
De Moivre's Theorem

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

41. When two complex numbers are multipiled together.
Complex Multiplication
Subfield
Rules of Complex Arithmetic
e^(ln z)

42. ½(e^(iz) + e^(-iz))
Roots of Unity
cos z
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - Multiplication by i

43. 1
(cos? +isin?)n
multiply the numerator and the denominator by the complex conjugate of the denominator.
multiplying complex numbers
i^4

44. Derives z = a+bi
How to find any Power
Irrational Number
Euler Formula
a + bi for some real a and b.

45. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
complex
Polar Coordinates - Division
non-integers

46. I = imaginary unit - i² = -1 or i = v-1
Imaginary Numbers
Absolute Value of a Complex Number
z - z*
How to add and subtract complex numbers (2-3i)-(4+6i)

47. (a + bi)(c + bi) =
Complex Exponentiation
We say that c+di and c-di are complex conjugates.
Polar Coordinates - Multiplication by i
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

48. 3rd. Rule of Complex Arithmetic
Complex Numbers: Multiply
(cos? +isin?)n
Polar Coordinates - cos?
For real a and b - a + bi = 0 if and only if a = b = 0

49. The reals are just the
i^0
conjugate pairs
z1 / z2
x-axis in the complex plane

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