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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. Given (4-2i) the complex conjugate would be (4+2i)
|z| = mod(z)
Field
Complex Division
Complex Conjugate

2. Rotates anticlockwise by p/2
conjugate pairs
i^1
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - Multiplication by i

3. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
the vector (a -b)
transcendental
z1 ^ (z2)

4. When two complex numbers are subtracted from one another.
Complex Subtraction
z + z*
0 if and only if a = b = 0
Subfield

5. The complex number z representing a+bi.
Complex Division
'i'
The Complex Numbers
Affix

6. V(x² + y²) = |z|
has a solution.
Polar Coordinates - r
Polar Coordinates - z
z1 / z2

7. Root negative - has letter i
e^(ln z)
Rational Number
Complex Multiplication
imaginary

8. The square root of -1.
subtracting complex numbers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z - z*
Imaginary Unit

9. Like pi
sin z
Real Numbers
Rational Number
transcendental

10. A subset within a field.
Subfield
e^(ln z)
How to multiply complex nubers(2+i)(2i-3)
v(-1)

11. I
|z-w|
Liouville's Theorem -
i^1
i^2 = -1

12. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
i^2 = -1
(a + bi) = (c + bi) = (a + c) + ( b + d)i
sin iy
conjugate

13. (e^(iz) - e^(-iz)) / 2i
Rational Number
the complex numbers
sin z
complex numbers

14. When two complex numbers are added together.
i^2
(cos? +isin?)n
Complex Addition
rational

15. Starts at 1 - does not include 0
Rational Number
|z-w|
natural
the complex numbers

16. 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
i^4
Rules of Complex Arithmetic
Irrational Number
z1 / z2

17. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
(cos? +isin?)n
Complex Multiplication
Field

18. xpressions such as ``the complex number z'' - and ``the point z'' are now
How to solve (2i+3)/(9-i)
Argand diagram
Complex Numbers: Multiply
interchangeable

19. 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
Square Root
z1 ^ (z2)
radicals

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

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21. E ^ (z2 ln z1)
Complex Numbers: Add & subtract
z1 ^ (z2)
How to solve (2i+3)/(9-i)
|z-w|

22. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
'i'
natural
Integers
(a + c) + ( b + d)i

23. 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.
Any polynomial O(xn) - (n > 0)
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Exponentiation
How to find any Power

24. All the powers of i can be written as
Rules of Complex Arithmetic
four different numbers: i - -i - 1 - and -1.
Complex Number
multiply the numerator and the denominator by the complex conjugate of the denominator.

25. x / r
radicals
i^2 = -1
Polar Coordinates - cos?
multiply the numerator and the denominator by the complex conjugate of the denominator.

26. y / r
The Complex Numbers
Polar Coordinates - sin?
has a solution.
How to multiply complex nubers(2+i)(2i-3)

27. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
Imaginary Numbers
Real Numbers
Complex Multiplication

28. 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.
real
complex numbers
standard form of complex numbers
Complex Numbers: Multiply

29. ? = -tan?
Polar Coordinates - Arg(z*)
Complex Number
i^3
Complex Division

30. Imaginary number

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31. Real and imaginary numbers
Polar Coordinates - Multiplication by i
Real and Imaginary Parts
real
complex numbers

32. V(zz*) = v(a² + b²)
Every complex number has the 'Standard Form': a + bi for some real a and b.
|z| = mod(z)
'i'
The Complex Numbers

33. Any number not rational
For real a and b - a + bi = 0 if and only if a = b = 0
i^0
i^4
irrational

34. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Polar Coordinates - cos?
Complex Conjugate
Imaginary Unit
Absolute Value of a Complex Number

35. Derives z = a+bi
Euler Formula
Imaginary Unit
De Moivre's Theorem
Imaginary Numbers

36. (2+i)(2i-3) you would use the foil methom which is first outter inner last. (2x2i)(2x-3)(ix2i^2)(ix(-3) =i-8
imaginary
Integers
How to multiply complex nubers(2+i)(2i-3)
conjugate pairs

37. (a + bi) = (c + bi) =
Complex Conjugate
interchangeable
(a + c) + ( b + d)i
x-axis in the complex plane

38. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Complex Conjugate
-1
multiplying complex numbers
interchangeable

39. Multiply moduli and add arguments
sin iy
Polar Coordinates - cos?
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - Multiplication

40. A number that cannot be expressed as a fraction for any integer.
Irrational Number
Affix
0 if and only if a = b = 0
z + z*

41. For real a and b - a + bi =
(a + bi) = (c + bi) = (a + c) + ( b + d)i
a real number: (a + bi)(a - bi) = a² + b²
0 if and only if a = b = 0
Complex Exponentiation

42. 1
i^0
ln z
Complex Numbers: Multiply
i^4

43. 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.
a real number: (a + bi)(a - bi) = a² + b²
Euler's Formula
Polar Coordinates - z?¹
Complex numbers are points in the plane

44. I
complex numbers
Imaginary number
i^4
v(-1)

45. 4th. Rule of Complex Arithmetic
integers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
(cos? +isin?)n
irrational

46. Where the curvature of the graph changes
cos iy
Imaginary number
ln z
point of inflection

47. Not on the numberline
natural
cos iy
non-integers
i^2

48. The modulus of the complex number z= a + ib now can be interpreted as
four different numbers: i - -i - 1 - and -1.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
i^0
the distance from z to the origin in the complex plane

49. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
Imaginary Numbers
sin iy
Roots of Unity

50. (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
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
How to solve (2i+3)/(9-i)
i²
-1