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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. Equivalent to an Imaginary Unit.
the complex numbers
Imaginary number
e^(ln z)
Euler's Formula

2. The product of an imaginary number and its conjugate is
z1 / z2
Absolute Value of a Complex Number
a real number: (a + bi)(a - bi) = a² + b²
conjugate pairs

3. 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
subtracting complex numbers
a + bi for some real a and b.
i^4
adding complex numbers

4. 3
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - sin?
Real Numbers
i^3

5. I
Roots of Unity
non-integers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
v(-1)

6. Like pi
How to solve (2i+3)/(9-i)
Irrational Number
Polar Coordinates - Division
transcendental

7. Root negative - has letter i
Polar Coordinates - Arg(z*)
imaginary
How to solve (2i+3)/(9-i)
Polar Coordinates - z

8. The square root of -1.
the complex numbers
How to multiply complex nubers(2+i)(2i-3)
Complex Exponentiation
Imaginary Unit

9. 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
has a solution.
Euler's Formula
Roots of Unity

10. x + iy = r(cos? + isin?) = re^(i?)
-1
Polar Coordinates - z
sin iy
0 if and only if a = b = 0

11. Divide moduli and subtract arguments
Polar Coordinates - Arg(z*)
Polar Coordinates - Division
the vector (a -b)
-1

12. (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
cos z
How to solve (2i+3)/(9-i)
Complex numbers are points in the plane
standard form of complex numbers

13. (a + bi) = (c + bi) =
four different numbers: i - -i - 1 - and -1.
v(-1)
i²
(a + c) + ( b + d)i

14. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
conjugate pairs
Polar Coordinates - Multiplication by i
complex numbers

15. 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
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - z
Argand diagram
We say that c+di and c-di are complex conjugates.

16. The modulus of the complex number z= a + ib now can be interpreted as
How to multiply complex nubers(2+i)(2i-3)
i²
the distance from z to the origin in the complex plane
standard form of complex numbers

17. (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
'i'
the distance from z to the origin in the complex plane
How to multiply complex nubers(2+i)(2i-3)
interchangeable

18. R?¹(cos? - isin?)
the vector (a -b)
Euler's Formula
point of inflection
Polar Coordinates - z?¹

19. 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
0 if and only if a = b = 0
Imaginary Numbers
Real and Imaginary Parts
For real a and b - a + bi = 0 if and only if a = b = 0

20. Numbers on a numberline
ln z
non-integers
sin iy
integers

21. All numbers
z1 / z2
complex
Complex Number
has a solution.

22. x / r
non-integers
Polar Coordinates - cos?
natural
Imaginary number

23. Imaginary number

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

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25. (e^(iz) - e^(-iz)) / 2i
How to multiply complex nubers(2+i)(2i-3)
Complex Multiplication
Polar Coordinates - Multiplication
sin z

26. Multiply moduli and add arguments
multiplying complex numbers
integers
Polar Coordinates - Multiplication
a real number: (a + bi)(a - bi) = a² + b²

27. A complex number may be taken to the power of another complex number.
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Exponentiation
cos z
Polar Coordinates - r

28. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Real Numbers
Absolute Value of a Complex Number
-1
v(-1)

29. For real a and b - a + bi =
Euler's Formula
the vector (a -b)
Affix
0 if and only if a = b = 0

30. Cos n? + i sin n? (for all n integers)
-1
cos iy
has a solution.
(cos? +isin?)n

31. When two complex numbers are multipiled together.
z1 ^ (z2)
e^(ln z)
Complex Multiplication
z + z*

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

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33. In this amazing number field every algebraic equation in z with complex coefficients
Rational Number
Polar Coordinates - Multiplication by i
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
has a solution.

34. Has exactly n roots by the fundamental theorem of algebra
a + bi for some real a and b.
Polar Coordinates - Multiplication
Any polynomial O(xn) - (n > 0)
complex numbers

35. A subset within a field.
standard form of complex numbers
Complex Exponentiation
Subfield
Polar Coordinates - sin?

36. To simplify a complex fraction
Complex Numbers: Multiply
-1
multiply the numerator and the denominator by the complex conjugate of the denominator.
How to find any Power

37. ½(e^(-y) +e^(y)) = cosh y
cos iy
|z| = mod(z)
The Complex Numbers
Irrational Number

38. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
How to solve (2i+3)/(9-i)
Polar Coordinates - cos?
non-integers

39. 2ib
Complex Conjugate
We say that c+di and c-di are complex conjugates.
z1 ^ (z2)
z - z*

40. 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.
cos iy
Complex Numbers: Multiply
De Moivre's Theorem
i^2 = -1

41. Real and imaginary numbers
Real and Imaginary Parts
complex numbers
The Complex Numbers
Polar Coordinates - Division

42. y / r
Polar Coordinates - sin?
integers
Complex numbers are points in the plane
Euler's Formula

43. 2nd. Rule of Complex Arithmetic

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44. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
i²
How to add and subtract complex numbers (2-3i)-(4+6i)
the distance from z to the origin in the complex plane
a + bi for some real a and b.

45. A² + b² - real and non negative
i^3
zz*
(cos? +isin?)n
point of inflection

46. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Real and Imaginary Parts
The Complex Numbers
Irrational Number
v(-1)

47. z1z2* / |z2|²
(cos? +isin?)n
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
-1
z1 / z2

48. 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.
We say that c+di and c-di are complex conjugates.
Complex numbers are points in the plane
non-integers
Polar Coordinates - z?¹

49. 1
How to add and subtract complex numbers (2-3i)-(4+6i)
multiply the numerator and the denominator by the complex conjugate of the denominator.
(a + c) + ( b + d)i
cosh²y - sinh²y

50. xpressions such as ``the complex number z'' - and ``the point z'' are now
Polar Coordinates - z
four different numbers: i - -i - 1 - and -1.
0 if and only if a = b = 0
interchangeable