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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. y / r
(cos? +isin?)n
i^1
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - sin?

2. 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.
i^4
Complex numbers are points in the plane
|z-w|
How to add and subtract complex numbers (2-3i)-(4+6i)

3. 3
z + z*
Every complex number has the 'Standard Form': a + bi for some real a and b.
i^3
'i'

4. A number that can be expressed as a fraction p/q where q is not equal to 0.
conjugate pairs
Rational Number
the vector (a -b)
Polar Coordinates - r

5. 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
four different numbers: i - -i - 1 - and -1.
adding complex numbers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Multiplication

6. Cos n? + i sin n? (for all n integers)
(a + c) + ( b + d)i
(cos? +isin?)n
Complex Numbers: Multiply
Irrational Number

7. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Polar Coordinates - Multiplication
i²
Complex Number
Integers

8. We can also think of the point z= a+ ib as
the vector (a -b)
Complex Addition
Argand diagram
Complex Multiplication

9. (e^(iz) - e^(-iz)) / 2i
Complex numbers are points in the plane
For real a and b - a + bi = 0 if and only if a = b = 0
sin z
rational

10. R^2 = x
Polar Coordinates - Multiplication
Euler Formula
Square Root
Complex numbers are points in the plane

11. When two complex numbers are divided.
Field
cosh²y - sinh²y
Complex Division
De Moivre's Theorem

12. Like pi
We say that c+di and c-di are complex conjugates.
0 if and only if a = b = 0
De Moivre's Theorem
transcendental

13. In this amazing number field every algebraic equation in z with complex coefficients
Field
has a solution.
Roots of Unity
ln z

14. Given (4-2i) the complex conjugate would be (4+2i)
rational
Complex Conjugate
natural
Polar Coordinates - Division

15. x + iy = r(cos? + isin?) = re^(i?)
Complex Number
Polar Coordinates - z
Polar Coordinates - Multiplication by i
point of inflection

16. R?¹(cos? - isin?)
sin iy
radicals
Real Numbers
Polar Coordinates - z?¹

17. A + bi
standard form of complex numbers
z1 / z2
interchangeable
i^4

18. I^2 =
(a + c) + ( b + d)i
i^2
-1
a real number: (a + bi)(a - bi) = a² + b²

19. (a + bi)(c + bi) =
Complex numbers are points in the plane
Subfield
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
adding complex numbers

20. Divide moduli and subtract arguments
the complex numbers
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Numbers: Multiply
Polar Coordinates - Division

21. 2a
i^1
z + z*
transcendental
i^3

22. ½(e^(-y) +e^(y)) = cosh y
ln z
Polar Coordinates - r
cos iy
integers

23. A number that cannot be expressed as a fraction for any integer.
z1 ^ (z2)
'i'
e^(ln z)
Irrational Number

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

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25. No i
Square Root
adding complex numbers
'i'
real

26. Not on the numberline
the distance from z to the origin in the complex plane
non-integers
Complex Numbers: Multiply
i^2

27. 2ib
z - z*
transcendental
Affix
sin z

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

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29. Written as fractions - terminating + repeating decimals
has a solution.
Irrational Number
interchangeable
rational

30. Real and imaginary numbers
cos iy
four different numbers: i - -i - 1 - and -1.
rational
complex numbers

31. (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
How to solve (2i+3)/(9-i)
Field
can't get out of the complex numbers by adding (or subtracting) or multiplying two
How to add and subtract complex numbers (2-3i)-(4+6i)

32. We see in this way that the distance between two points z and w in the complex plane is
We say that c+di and c-di are complex conjugates.
|z-w|
(cos? +isin?)n
v(-1)

33. To simplify a complex fraction
i²
the distance from z to the origin in the complex plane
multiply the numerator and the denominator by the complex conjugate of the denominator.
irrational

34. A complex number and its conjugate
conjugate pairs
i^3
Polar Coordinates - Arg(z*)
Complex Numbers: Add & subtract

35. I^26/4= i^24 x i^2 =-1 so u divide the number by 4 and whatevers left over is the number that its equal to.
Complex numbers are points in the plane
i^2 = -1
How to find any Power
a real number: (a + bi)(a - bi) = a² + b²

36. The modulus of the complex number z= a + ib now can be interpreted as
The Complex Numbers
i^4
x-axis in the complex plane
the distance from z to the origin in the complex plane

37. All numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.
real
Imaginary Numbers
complex

38. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
De Moivre's Theorem
z1 ^ (z2)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
multiplying complex numbers

39. To simplify the square root of a negative number
z + z*
How to multiply complex nubers(2+i)(2i-3)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Affix

40. (a + bi) = (c + bi) =
subtracting complex numbers
(a + c) + ( b + d)i
multiplying complex numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i

41. 1
adding complex numbers
Roots of Unity
i^0
integers

42. Imaginary number

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43. All the powers of i can be written as
Every complex number has the 'Standard Form': a + bi for some real a and b.
Imaginary Unit
sin z
four different numbers: i - -i - 1 - and -1.

44. 1
multiply the numerator and the denominator by the complex conjugate of the denominator.
i^4
(a + bi) = (c + bi) = (a + c) + ( b + d)i
sin z

45. Derives z = a+bi
Euler Formula
adding complex numbers
i^2
Polar Coordinates - Arg(z*)

46. Numbers on a numberline
radicals
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
integers
Argand diagram

47. 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
How to add and subtract complex numbers (2-3i)-(4+6i)
a real number: (a + bi)(a - bi) = a² + b²
conjugate

48. E ^ (z2 ln z1)
We say that c+di and c-di are complex conjugates.
ln z
Every complex number has the 'Standard Form': a + bi for some real a and b.
z1 ^ (z2)

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
interchangeable
Real and Imaginary Parts
subtracting complex numbers
-1

50. The product of an imaginary number and its conjugate is
Roots of Unity
a real number: (a + bi)(a - bi) = a² + b²
De Moivre's Theorem
0 if and only if a = b = 0