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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. When two complex numbers are multipiled together.
Complex Multiplication
Complex Subtraction
-1
z + z*

2. x / r
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - cos?
radicals
multiplying complex numbers

3. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
non-integers
imaginary
-1
multiplying complex numbers

4. I = imaginary unit - i² = -1 or i = v-1
Complex Addition
Imaginary Numbers
For real a and b - a + bi = 0 if and only if a = b = 0
interchangeable

5. We see in this way that the distance between two points z and w in the complex plane is
Polar Coordinates - Division
Complex Conjugate
|z-w|
a real number: (a + bi)(a - bi) = a² + b²

6. E^(ln r) e^(i?) e^(2pin)
Affix
Complex Numbers: Add & subtract
i^1
e^(ln z)

7. The complex number z representing a+bi.
the complex numbers
radicals
Affix
non-integers

8. ? = -tan?
the distance from z to the origin in the complex plane
Real Numbers
Complex Addition
Polar Coordinates - Arg(z*)

9. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
multiplying complex numbers
-1
integers

10. 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
z + z*
Liouville's Theorem -
Integers

11. For real a and b - a + bi =
i^2 = -1
0 if and only if a = b = 0
interchangeable
Polar Coordinates - Division

12. The modulus of the complex number z= a + ib now can be interpreted as
'i'
rational
zz*
the distance from z to the origin in the complex plane

13. The product of an imaginary number and its conjugate is
Complex Number
non-integers
a real number: (a + bi)(a - bi) = a² + b²
How to solve (2i+3)/(9-i)

14. No i
interchangeable
real
Real Numbers
Affix

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
We say that c+di and c-di are complex conjugates.
Complex Division
four different numbers: i - -i - 1 - and -1.
Complex Numbers: Add & subtract

16. Derives z = a+bi
Euler Formula
natural
Polar Coordinates - sin?
0 if and only if a = b = 0

17. 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
The Complex Numbers
Field
Real and Imaginary Parts
v(-1)

18. V(x² + y²) = |z|
z1 / z2
e^(ln z)
Polar Coordinates - r
cos iy

19. Given (4-2i) the complex conjugate would be (4+2i)
Complex Exponentiation
Roots of Unity
Complex Conjugate
imaginary

20. Notice that rules 4 and 5 state that we complex numbers together - we can divide by c + di if c and d are not both zero. But there is a much easier way to do division.

21. A number that cannot be expressed as a fraction for any integer.
How to find any Power
cos z
Any polynomial O(xn) - (n > 0)
Irrational Number

22. Numbers on a numberline
cos z
integers
Polar Coordinates - z?¹
i^3

23. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
Euler's Formula
We say that c+di and c-di are complex conjugates.
integers

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

25. 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: Multiply
How to find any Power
the complex numbers
Real and Imaginary Parts

26. Not on the numberline
non-integers
How to multiply complex nubers(2+i)(2i-3)
Complex Numbers: Add & subtract
radicals

27. R^2 = x
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - Division
Square Root
the vector (a -b)

28. 1
can't get out of the complex numbers by adding (or subtracting) or multiplying two
i²
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
x-axis in the complex plane

29. Any number not rational
i^2 = -1
Complex Exponentiation
|z-w|
irrational

30. Has exactly n roots by the fundamental theorem of algebra
the vector (a -b)
How to multiply complex nubers(2+i)(2i-3)
Imaginary Numbers
Any polynomial O(xn) - (n > 0)

31. When two complex numbers are divided.
Polar Coordinates - z?¹
Complex Division
i^2 = -1
cos iy

32. 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.
Complex numbers are points in the plane
irrational
complex
i^3

33. A subset within a field.
Polar Coordinates - z?¹
Subfield
Roots of Unity
Complex Numbers: Add & subtract

34. Divide moduli and subtract arguments
Real Numbers
|z-w|
Polar Coordinates - Division
interchangeable

35. 2nd. Rule of Complex Arithmetic

36. 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
z + z*
Polar Coordinates - cos?
zz*
adding complex numbers

37. 1st. Rule of Complex Arithmetic
transcendental
i^2 = -1
Imaginary Unit
|z| = mod(z)

38. A complex number may be taken to the power of another complex number.
subtracting complex numbers
Polar Coordinates - Arg(z*)
Complex Exponentiation
irrational

39. 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
Absolute Value of a Complex Number
For real a and b - a + bi = 0 if and only if a = b = 0
Field

40. The square root of -1.
(cos? +isin?)n
cos iy
Complex Multiplication
Imaginary Unit

41. x + iy = r(cos? + isin?) = re^(i?)
cosh²y - sinh²y
'i'
Polar Coordinates - z
Complex Multiplication

42. 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.
non-integers
v(-1)
Polar Coordinates - Arg(z*)
Complex Numbers: Multiply

43. Imaginary number

44. 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
Rational Number
subtracting complex numbers
We say that c+di and c-di are complex conjugates.
Polar Coordinates - Division

45. I
Polar Coordinates - Multiplication by i
rational
Complex Multiplication
v(-1)

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 Numbers
multiplying complex numbers
The Complex Numbers
Any polynomial O(xn) - (n > 0)

47. 1
i^2
0 if and only if a = b = 0
i^2 = -1
How to find any Power

48. A number that can be expressed as a fraction p/q where q is not equal to 0.
The Complex Numbers
Rational Number
x-axis in the complex plane
(a + bi) = (c + bi) = (a + c) + ( b + d)i

49. A complex number and its conjugate
Irrational Number
i^1
z - z*
conjugate pairs

50. (e^(iz) - e^(-iz)) / 2i
Polar Coordinates - Division
-1
a + bi for some real a and b.
sin z