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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.
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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. We see in this way that the distance between two points z and w in the complex plane is
Complex Exponentiation
|z-w|
Real and Imaginary Parts
ln z

2. 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.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Numbers: Multiply
Euler's Formula
integers

3. We can also think of the point z= a+ ib as
imaginary
sin iy
radicals
the vector (a -b)

4. 3
Rational Number
i^3
Subfield
|z| = mod(z)

5. Root negative - has letter i
a + bi for some real a and b.
imaginary
conjugate
We say that c+di and c-di are complex conjugates.

6. The modulus of the complex number z= a + ib now can be interpreted as
i²
the distance from z to the origin in the complex plane
e^(ln z)
adding complex numbers

7. A + bi
imaginary
standard form of complex numbers
multiplying complex numbers
real

8. I
Imaginary number
v(-1)
Polar Coordinates - Division
Polar Coordinates - sin?

9. The complex number z representing a+bi.
Affix
the vector (a -b)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
multiply the numerator and the denominator by the complex conjugate of the denominator.

10. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
i²
multiplying complex numbers
How to find any Power

11. 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
Real and Imaginary Parts
Polar Coordinates - Multiplication
complex numbers

12. 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
interchangeable
adding complex numbers
Affix
can't get out of the complex numbers by adding (or subtracting) or multiplying two

13. E ^ (z2 ln z1)
z1 ^ (z2)
Polar Coordinates - sin?
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Euler Formula

14. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Integers
Rules of Complex Arithmetic
adding complex numbers
Irrational Number

15. All numbers
i^3
Complex Conjugate
Polar Coordinates - cos?
complex

16. x + iy = r(cos? + isin?) = re^(i?)
-1
Polar Coordinates - z
sin iy
Polar Coordinates - Arg(z*)

17. ½(e^(iz) + e^(-iz))
Liouville's Theorem -
cos z
Roots of Unity
point of inflection

18. A+bi
Polar Coordinates - sin?
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Number Formula
Field

19. Cos n? + i sin n? (for all n integers)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Argand diagram
(a + c) + ( b + d)i
(cos? +isin?)n

20. R?¹(cos? - isin?)
Polar Coordinates - Multiplication
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - z?¹
|z-w|

21. (e^(iz) - e^(-iz)) / 2i
natural
(a + c) + ( b + d)i
|z-w|
sin z

22. 5th. Rule of Complex Arithmetic
How to find any Power
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
the vector (a -b)
zz*

23. A plot of complex numbers as points.
Complex Division
Complex numbers are points in the plane
Roots of Unity
Argand diagram

24. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
Polar Coordinates - z
multiply the numerator and the denominator by the complex conjugate of the denominator.
multiplying complex numbers

25. A number that cannot be expressed as a fraction for any integer.
How to multiply complex nubers(2+i)(2i-3)
Irrational Number
cos iy
non-integers

26. z1z2* / |z2|²
Polar Coordinates - Arg(z*)
Irrational Number
z1 / z2
z1 ^ (z2)

27. Not on the numberline
i^3
the complex numbers
conjugate
non-integers

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

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29. Written as fractions - terminating + repeating decimals
rational
How to add and subtract complex numbers (2-3i)-(4+6i)
0 if and only if a = b = 0
Subfield

30. 2nd. Rule of Complex Arithmetic

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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
point of inflection
non-integers
How to solve (2i+3)/(9-i)
sin iy

32. (e^(-y) - e^(y)) / 2i = i sinh y
the distance from z to the origin in the complex plane
sin iy
Imaginary number
multiply the numerator and the denominator by the complex conjugate of the denominator.

33. Every complex number has the 'Standard Form':
|z| = mod(z)
Imaginary Numbers
Complex Conjugate
a + bi for some real a and b.

34. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
a + bi for some real a and b.
i^4
Complex Addition

35. All the powers of i can be written as
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Addition
four different numbers: i - -i - 1 - and -1.
integers

36. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
cos z
multiplying complex numbers
Field
Every complex number has the 'Standard Form': a + bi for some real a and b.

37. 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
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Euler Formula

38. xpressions such as ``the complex number z'' - and ``the point z'' are now
Every complex number has the 'Standard Form': a + bi for some real a and b.
Complex Number
interchangeable
Integers

39. 4th. Rule of Complex Arithmetic
sin iy
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - Multiplication by i
(a + bi) = (c + bi) = (a + c) + ( b + d)i

40. To simplify a complex fraction
real
complex
conjugate
multiply the numerator and the denominator by the complex conjugate of the denominator.

41. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
i^1
multiply the numerator and the denominator by the complex conjugate of the denominator.
v(-1)

42. y / r
Complex numbers are points in the plane
Polar Coordinates - sin?
Imaginary Numbers
Any polynomial O(xn) - (n > 0)

43. Real and imaginary numbers
i²
Complex Subtraction
has a solution.
complex numbers

44. 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
cos z
Complex Addition
Real and Imaginary Parts
the vector (a -b)

45. V(zz*) = v(a² + b²)
|z| = mod(z)
i^2 = -1
'i'
Complex numbers are points in the plane

46. Multiply moduli and add arguments
i^4
Liouville's Theorem -
i^0
Polar Coordinates - Multiplication

47. The field of all rational and irrational numbers.
|z-w|
Real Numbers
Polar Coordinates - cos?
Imaginary Numbers

48. A² + b² - real and non negative
zz*
z - z*
Complex Conjugate
point of inflection

49. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
z + z*
non-integers
(cos? +isin?)n

50. Numbers on a numberline
complex
Argand diagram
integers
Complex Division