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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. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - cos?
How to add and subtract complex numbers (2-3i)-(4+6i)
e^(ln z)

2. 2ib
Polar Coordinates - Arg(z*)
z - z*
Liouville's Theorem -
point of inflection

3. E^(ln r) e^(i?) e^(2pin)
How to multiply complex nubers(2+i)(2i-3)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
e^(ln z)
'i'

4. 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.
i^0
sin z
Imaginary Numbers
How to find any Power

5. 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.

6. The product of an imaginary number and its conjugate is
the complex numbers
complex
Polar Coordinates - Division
a real number: (a + bi)(a - bi) = a² + b²

7. 2a
z + z*
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Exponentiation
i^3

8. Rotates anticlockwise by p/2
Complex Division
Imaginary Numbers
Polar Coordinates - Multiplication by i
How to multiply complex nubers(2+i)(2i-3)

9. The modulus of the complex number z= a + ib now can be interpreted as
the distance from z to the origin in the complex plane
Real Numbers
Complex Division
Absolute Value of a Complex Number

10. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Polar Coordinates - Multiplication
|z-w|
The Complex Numbers
How to multiply complex nubers(2+i)(2i-3)

11. V(zz*) = v(a² + b²)
Complex numbers are points in the plane
Argand diagram
imaginary
|z| = mod(z)

12. ½(e^(iz) + e^(-iz))
cos z
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to multiply complex nubers(2+i)(2i-3)
Rules of Complex Arithmetic

13. A² + b² - real and non negative
the complex numbers
Argand diagram
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
zz*

14. Equivalent to an Imaginary Unit.
(cos? +isin?)n
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Imaginary number
multiplying complex numbers

15. 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.
Irrational Number
z + z*
Complex numbers are points in the plane
(a + c) + ( b + d)i

16. Given (4-2i) the complex conjugate would be (4+2i)
Imaginary Unit
multiplying complex numbers
Complex Conjugate
point of inflection

17. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Liouville's Theorem -
Complex Addition
Polar Coordinates - Multiplication
conjugate

18. The field of numbers of the form - where and are real numbers and i is the imaginary unit equal to the square root of - . When a single letter is used to denote a complex number - it is sometimes called an 'affix.'
adding complex numbers
Complex Number
x-axis in the complex plane
(cos? +isin?)n

19. 1
a + bi for some real a and b.
i^4
ln z
has a solution.

20. R^2 = x
Liouville's Theorem -
can't get out of the complex numbers by adding (or subtracting) or multiplying two
i^2
Square Root

21. 5th. Rule of Complex Arithmetic
z1 ^ (z2)
Polar Coordinates - z?¹
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
conjugate

22. Divide moduli and subtract arguments
conjugate
Polar Coordinates - Division
Imaginary Numbers
Imaginary Unit

23. I = imaginary unit - i² = -1 or i = v-1
irrational
Real Numbers
Irrational Number
Imaginary Numbers

24. When two complex numbers are added together.
Argand diagram
Complex Addition
Integers
Complex Multiplication

25. A complex number and its conjugate
De Moivre's Theorem
z1 ^ (z2)
Euler's Formula
conjugate pairs

26. We see in this way that the distance between two points z and w in the complex plane is
Complex Numbers: Multiply
|z-w|
a + bi for some real a and b.
conjugate

27. No i
conjugate pairs
zz*
(cos? +isin?)n
real

28. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Complex Exponentiation
Absolute Value of a Complex Number
Complex Division
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

29. When two complex numbers are divided.
i^2 = -1
the distance from z to the origin in the complex plane
0 if and only if a = b = 0
Complex Division

30. 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.
Every complex number has the 'Standard Form': a + bi for some real a and b.
Imaginary number
Complex Numbers: Multiply
Polar Coordinates - cos?

31. y / r
Polar Coordinates - sin?
Complex Number Formula
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
adding complex numbers

32. I
i^1
Complex Conjugate
non-integers
four different numbers: i - -i - 1 - and -1.

33. Where the curvature of the graph changes
i^3
point of inflection
e^(ln z)
sin iy

34. A plot of complex numbers as points.
irrational
Complex Subtraction
Polar Coordinates - r
Argand diagram

35. The square root of -1.
i^2
sin iy
multiplying complex numbers
Imaginary Unit

36. E ^ (z2 ln z1)
Imaginary Numbers
Any polynomial O(xn) - (n > 0)
Euler Formula
z1 ^ (z2)

37. 1
Any polynomial O(xn) - (n > 0)
i^0
Rules of Complex Arithmetic
has a solution.

38. (a + bi)(c + bi) =
e^(ln z)
a + bi for some real a and b.
natural
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

39. Derives z = a+bi
Rational Number
'i'
Euler Formula
The Complex Numbers

40. Has exactly n roots by the fundamental theorem of algebra
Rational Number
Any polynomial O(xn) - (n > 0)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
zz*

41. R?¹(cos? - isin?)
cosh²y - sinh²y
Complex Numbers: Add & subtract
Polar Coordinates - r
Polar Coordinates - z?¹

42. 3rd. Rule of Complex Arithmetic
Irrational Number
Argand diagram
For real a and b - a + bi = 0 if and only if a = b = 0
i^0

43. 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
z - z*
z1 / z2
(a + bi) = (c + bi) = (a + c) + ( b + d)i

44. To simplify a complex fraction
z - z*
Complex Multiplication
multiply the numerator and the denominator by the complex conjugate of the denominator.
conjugate pairs

45. A complex number may be taken to the power of another complex number.
Complex Exponentiation
z - z*
can't get out of the complex numbers by adding (or subtracting) or multiplying two
four different numbers: i - -i - 1 - and -1.

46. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
irrational
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Division
multiplying complex numbers

47. 2nd. Rule of Complex Arithmetic

48. 1
i²
Complex Exponentiation
De Moivre's Theorem
i^3

49. 1
i^2
i^1
How to multiply complex nubers(2+i)(2i-3)
Rules of Complex Arithmetic

50. Real and imaginary numbers
Polar Coordinates - Multiplication by i
complex numbers
Complex Numbers: Multiply
Complex Numbers: Add & subtract