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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
transcendental
Complex Subtraction
i²
i^2

2. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
De Moivre's Theorem
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Field

3. Real and imaginary numbers
complex numbers
x-axis in the complex plane
How to solve (2i+3)/(9-i)
transcendental

4. The reals are just the
Complex Numbers: Add & subtract
zz*
x-axis in the complex plane
subtracting complex numbers

5. All numbers
z + z*
Imaginary number
complex
conjugate pairs

6. A complex number and its conjugate
conjugate pairs
Complex Number Formula
z1 ^ (z2)
a + bi for some real a and b.

7. 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
Absolute Value of a Complex Number
Polar Coordinates - z?¹
Polar Coordinates - cos?

8. Rotates anticlockwise by p/2
a + bi for some real a and b.
complex numbers
Polar Coordinates - Multiplication by i
subtracting complex numbers

9. Every complex number has the 'Standard Form':
ln z
x-axis in the complex plane
a + bi for some real a and b.
i^3

10. To simplify a complex fraction
Square Root
x-axis in the complex plane
multiply the numerator and the denominator by the complex conjugate of the denominator.
cos iy

11. V(x² + y²) = |z|
Imaginary Unit
Polar Coordinates - r
Polar Coordinates - Multiplication
cosh²y - sinh²y

12. When two complex numbers are subtracted from one another.
Complex Subtraction
Rational Number
i^3
|z-w|

13. V(zz*) = v(a² + b²)
ln z
|z| = mod(z)
complex
Argand diagram

14. When two complex numbers are multipiled together.
Polar Coordinates - r
Complex Multiplication
Argand diagram
Field

15. Not on the numberline
non-integers
interchangeable
i^2 = -1
Polar Coordinates - z?¹

16. 3
Polar Coordinates - Multiplication
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^3
De Moivre's Theorem

17. 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
Polar Coordinates - Arg(z*)
Polar Coordinates - sin?
has a solution.

18. Derives z = a+bi
Euler Formula
Polar Coordinates - z
ln z
x-axis in the complex plane

19. In this amazing number field every algebraic equation in z with complex coefficients
complex numbers
a real number: (a + bi)(a - bi) = a² + b²
has a solution.
i^2

20. R^2 = x
Square Root
Imaginary Unit
integers
The Complex Numbers

21. y / r
0 if and only if a = b = 0
Roots of Unity
zz*
Polar Coordinates - sin?

22. 5th. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Addition
Complex Number

23. ½(e^(iz) + e^(-iz))
irrational
(cos? +isin?)n
four different numbers: i - -i - 1 - and -1.
cos z

24. The product of an imaginary number and its conjugate is
z + z*
De Moivre's Theorem
a real number: (a + bi)(a - bi) = a² + b²
|z-w|

25. All the powers of i can be written as
sin z
cos iy
four different numbers: i - -i - 1 - and -1.
Irrational Number

26. 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
Polar Coordinates - z?¹
Polar Coordinates - Division
Rules of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

27. A complex number may be taken to the power of another complex number.
Complex Exponentiation
Polar Coordinates - Multiplication by i
How to multiply complex nubers(2+i)(2i-3)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

28. Imaginary number

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29. 1
Real Numbers
i^0
i^3
Imaginary Numbers

30. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
has a solution.
cos iy
Complex Exponentiation
conjugate

31. 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.

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32. Any number not rational
'i'
can't get out of the complex numbers by adding (or subtracting) or multiplying two
irrational
adding complex numbers

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

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34. I = imaginary unit - i² = -1 or i = v-1
multiplying complex numbers
Imaginary Numbers
How to multiply complex nubers(2+i)(2i-3)
For real a and b - a + bi = 0 if and only if a = b = 0

35. Where the curvature of the graph changes
point of inflection
Roots of Unity
imaginary
multiplying complex numbers

36. The complex number z representing a+bi.
Real Numbers
Imaginary number
Affix
i^2 = -1

37. 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
i²
the vector (a -b)
the complex numbers
Complex Addition

38. When two complex numbers are divided.
Complex Division
conjugate
the distance from z to the origin in the complex plane
Absolute Value of a Complex Number

39. To simplify the square root of a negative number
Polar Coordinates - z
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
cos z
For real a and b - a + bi = 0 if and only if a = b = 0

40. Numbers on a numberline
a real number: (a + bi)(a - bi) = a² + b²
i^2 = -1
Complex Exponentiation
integers

41. A² + b² - real and non negative
natural
zz*
Affix
0 if and only if a = b = 0

42. A + bi
|z-w|
Field
standard form of complex numbers
0 if and only if a = b = 0

43. I
How to multiply complex nubers(2+i)(2i-3)
v(-1)
Complex Numbers: Multiply
z1 ^ (z2)

44. E^(ln r) e^(i?) e^(2pin)
Polar Coordinates - z?¹
can't get out of the complex numbers by adding (or subtracting) or multiplying two
e^(ln z)
-1

45. (a + bi)(c + bi) =
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
The Complex Numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

46. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
The Complex Numbers
adding complex numbers
z + z*
Real and Imaginary Parts

47. x + iy = r(cos? + isin?) = re^(i?)
irrational
i^1
Polar Coordinates - z
Complex Numbers: Multiply

48. 2ib
Polar Coordinates - Arg(z*)
Roots of Unity
-1
z - z*

49. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Imaginary number
Euler's Formula
Field
i^0

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

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