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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. 1
How to add and subtract complex numbers (2-3i)-(4+6i)
Subfield
i^0
i^2

2. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Irrational Number
radicals
Absolute Value of a Complex Number
z - z*

3. ½(e^(-y) +e^(y)) = cosh y
cos iy
Square Root
Complex Number
subtracting complex numbers

4. 1
i^0
Polar Coordinates - sin?
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Number Formula

5. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
How to solve (2i+3)/(9-i)
conjugate pairs
sin iy

6. I
How to solve (2i+3)/(9-i)
Argand diagram
v(-1)
real

7. 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.
How to find any Power
Imaginary Unit
cosh²y - sinh²y
Euler Formula

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

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9. A number that can be expressed as a fraction p/q where q is not equal to 0.
De Moivre's Theorem
multiplying complex numbers
We say that c+di and c-di are complex conjugates.
Rational Number

10. Numbers on a numberline
Complex Number Formula
integers
subtracting complex numbers
Imaginary Unit

11. The square root of -1.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Imaginary Unit
Any polynomial O(xn) - (n > 0)
sin z

12. A complex number and its conjugate
conjugate pairs
How to multiply complex nubers(2+i)(2i-3)
Rules of Complex Arithmetic
Euler's Formula

13. R^2 = x
We say that c+di and c-di are complex conjugates.
Field
Square Root
z - z*

14. For real a and b - a + bi =
a real number: (a + bi)(a - bi) = a² + b²
The Complex Numbers
0 if and only if a = b = 0
|z| = mod(z)

15. 3rd. Rule of Complex Arithmetic
Real Numbers
For real a and b - a + bi = 0 if and only if a = b = 0
Argand diagram
Field

16. The reals are just the
i^3
x-axis in the complex plane
Absolute Value of a Complex Number
Polar Coordinates - sin?

17. Written as fractions - terminating + repeating decimals
Rules of Complex Arithmetic
rational
Imaginary Unit
e^(ln z)

18. x / r
Real Numbers
Polar Coordinates - cos?
|z| = mod(z)
'i'

19. (a + bi) = (c + bi) =
subtracting complex numbers
ln z
(a + c) + ( b + d)i
standard form of complex numbers

20. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
the complex numbers
Roots of Unity
the distance from z to the origin in the complex plane
x-axis in the complex plane

21. No i
non-integers
real
Polar Coordinates - Multiplication by i
has a solution.

22. 1st. Rule of Complex Arithmetic
adding complex numbers
Complex Number Formula
i^2 = -1
cos iy

23. The complex number z representing a+bi.
Complex Exponentiation
Absolute Value of a Complex Number
Affix
z1 ^ (z2)

24. Derives z = a+bi
Euler Formula
integers
The Complex Numbers
|z| = mod(z)

25. All the powers of i can be written as
|z| = mod(z)
the distance from z to the origin in the complex plane
can't get out of the complex numbers by adding (or subtracting) or multiplying two
four different numbers: i - -i - 1 - and -1.

26. We can also think of the point z= a+ ib as
Complex Number
v(-1)
Complex Numbers: Multiply
the vector (a -b)

27. Multiply moduli and add arguments
Imaginary Unit
Polar Coordinates - Multiplication
Any polynomial O(xn) - (n > 0)
can't get out of the complex numbers by adding (or subtracting) or multiplying two

28. A + bi
conjugate pairs
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
sin z
standard form of complex numbers

29. Given (4-2i) the complex conjugate would be (4+2i)
Integers
Complex Conjugate
sin z
Rules of Complex Arithmetic

30. 1
the complex numbers
cosh²y - sinh²y
We say that c+di and c-di are complex conjugates.
|z-w|

31. 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
Real Numbers
integers
Rules of Complex Arithmetic
irrational

32. 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
sin z
complex
Complex Subtraction
adding complex numbers

33. Divide moduli and subtract arguments
z1 ^ (z2)
Polar Coordinates - z
Polar Coordinates - Division
a + bi for some real a and b.

34. (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
Complex Numbers: Multiply
a + bi for some real a and b.
How to solve (2i+3)/(9-i)
How to multiply complex nubers(2+i)(2i-3)

35. To simplify the square root of a negative number
i^4
Roots of Unity
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
z + z*

36. Any number not rational
Polar Coordinates - r
irrational
(a + bi) = (c + bi) = (a + c) + ( b + d)i
|z-w|

37. Like pi
Rules of Complex Arithmetic
multiply the numerator and the denominator by the complex conjugate of the denominator.
transcendental
ln z

38. Have radical
radicals
(a + c) + ( b + d)i
Imaginary Numbers
transcendental

39. 2nd. Rule of Complex Arithmetic

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40. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
interchangeable
ln z
The Complex Numbers
Complex Number Formula

41. ? = -tan?
Polar Coordinates - Arg(z*)
Imaginary Unit
complex
The Complex Numbers

42. Real and imaginary numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.
Complex Subtraction
complex numbers
sin iy

43. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
How to solve (2i+3)/(9-i)
sin iy
Polar Coordinates - Multiplication

44. ½(e^(iz) + e^(-iz))
point of inflection
transcendental
cos z
Complex Exponentiation

45. 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.
zz*
How to find any Power
natural
Complex Numbers: Multiply

46. y / r
Complex Exponentiation
De Moivre's Theorem
i^2
Polar Coordinates - sin?

47. A+bi
i^0
(a + c) + ( b + d)i
Complex Number Formula
Complex Multiplication

48. (e^(iz) - e^(-iz)) / 2i
point of inflection
subtracting complex numbers
sin z
Complex Subtraction

49. A² + b² - real and non negative
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Any polynomial O(xn) - (n > 0)
Complex Addition
zz*

50. I = imaginary unit - i² = -1 or i = v-1
-1
Every complex number has the 'Standard Form': a + bi for some real a and b.
z1 / z2
Imaginary Numbers