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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
|z-w|
Subfield
How to add and subtract complex numbers (2-3i)-(4+6i)
(a + c) + ( b + d)i

2. 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 vector (a -b)
Polar Coordinates - Multiplication
zz*
Real and Imaginary Parts

3. R^2 = x
Square Root
z - z*
Complex Multiplication
How to multiply complex nubers(2+i)(2i-3)

4. All the powers of i can be written as
multiplying complex numbers
four different numbers: i - -i - 1 - and -1.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
a + bi for some real a and b.

5. 1
i^4
How to solve (2i+3)/(9-i)
How to find any Power
(a + c) + ( b + d)i

6. Divide moduli and subtract arguments
Complex Addition
Complex Multiplication
Polar Coordinates - Division
cosh²y - sinh²y

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

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8. A plot of complex numbers as points.
(cos? +isin?)n
Real and Imaginary Parts
Polar Coordinates - Multiplication
Argand diagram

9. The complex number z representing a+bi.
Complex Addition
For real a and b - a + bi = 0 if and only if a = b = 0
Affix
Liouville's Theorem -

10. 2nd. Rule of Complex Arithmetic

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11. A + bi
For real a and b - a + bi = 0 if and only if a = b = 0
standard form of complex numbers
Field
Complex Multiplication

12. Like pi
a + bi for some real a and b.
cos iy
transcendental
Affix

13. 1
Subfield
point of inflection
transcendental
i^2

14. For real a and b - a + bi =
a real number: (a + bi)(a - bi) = a² + b²
0 if and only if a = b = 0
v(-1)
Complex Numbers: Add & subtract

15. Written as fractions - terminating + repeating decimals
z + z*
rational
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Numbers: Add & subtract

16. The modulus of the complex number z= a + ib now can be interpreted as
(a + c) + ( b + d)i
the distance from z to the origin in the complex plane
Imaginary Unit
i^2

17. V(zz*) = v(a² + b²)
Imaginary number
|z| = mod(z)
De Moivre's Theorem
'i'

18. A number that cannot be expressed as a fraction for any integer.
How to add and subtract complex numbers (2-3i)-(4+6i)
Irrational Number
Square Root
How to solve (2i+3)/(9-i)

19. 4th. Rule of Complex Arithmetic
cosh²y - sinh²y
(a + bi) = (c + bi) = (a + c) + ( b + d)i
cos z
subtracting complex numbers

20. The field of all rational and irrational numbers.
conjugate pairs
Real Numbers
We say that c+di and c-di are complex conjugates.
Polar Coordinates - Division

21. When two complex numbers are multipiled together.
transcendental
cos z
Complex Multiplication
conjugate pairs

22. 1
Roots of Unity
the vector (a -b)
Square Root
cosh²y - sinh²y

23. 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
Rules of Complex Arithmetic
z1 / z2
Polar Coordinates - Division
sin z

24. 1
How to add and subtract complex numbers (2-3i)-(4+6i)
The Complex Numbers
z1 ^ (z2)
i²

25. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
Rational Number
Complex Number
complex

26. ? = -tan?
Polar Coordinates - Arg(z*)
can't get out of the complex numbers by adding (or subtracting) or multiplying two
interchangeable
Euler Formula

27. 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
Complex Addition
De Moivre's Theorem
Polar Coordinates - z?¹
subtracting complex numbers

28. Starts at 1 - does not include 0
Complex Subtraction
Polar Coordinates - r
Polar Coordinates - sin?
natural

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

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

31. 3
i^2
(a + bi) = (c + bi) = (a + c) + ( b + d)i
a + bi for some real a and b.
i^3

32. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
i^2 = -1
e^(ln z)
Complex Numbers: Multiply
Field

33. A number that can be expressed as a fraction p/q where q is not equal to 0.
the vector (a -b)
Complex Numbers: Multiply
interchangeable
Rational Number

34. We can also think of the point z= a+ ib as
i^4
|z-w|
complex numbers
the vector (a -b)

35. 3rd. Rule of Complex Arithmetic
Square Root
For real a and b - a + bi = 0 if and only if a = b = 0
standard form of complex numbers
Polar Coordinates - z

36. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
the vector (a -b)
Roots of Unity
Complex Subtraction

37. y / r
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Subtraction
Polar Coordinates - sin?
v(-1)

38. 1
i^0
Liouville's Theorem -
Complex Division
How to multiply complex nubers(2+i)(2i-3)

39. Not on the numberline
z1 ^ (z2)
Argand diagram
|z-w|
non-integers

40. 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
Subfield
Complex Numbers: Add & subtract
the complex numbers
subtracting complex numbers

41. To prove that number field every algebraic equation in z with complex coefficients has a solution we need

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42. (a + bi)(c + bi) =
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
four different numbers: i - -i - 1 - and -1.
Rational Number
Absolute Value of a Complex Number

43. When two complex numbers are subtracted from one another.
(a + c) + ( b + d)i
e^(ln z)
Complex Subtraction
z1 / z2

44. No i
0 if and only if a = b = 0
real
(cos? +isin?)n
Every complex number has the 'Standard Form': a + bi for some real a and b.

45. To simplify the square root of a negative number
Every complex number has the 'Standard Form': a + bi for some real a and b.
How to add and subtract complex numbers (2-3i)-(4+6i)
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
radicals

46. Given (4-2i) the complex conjugate would be (4+2i)
four different numbers: i - -i - 1 - and -1.
natural
Complex Conjugate
|z-w|

47. Where the curvature of the graph changes
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to add and subtract complex numbers (2-3i)-(4+6i)
point of inflection
z1 / z2

48. I = imaginary unit - i² = -1 or i = v-1
i^1
Complex Addition
|z| = mod(z)
Imaginary Numbers

49. I
v(-1)
i^1
Complex Division
adding complex numbers

50. Have radical
Subfield
Polar Coordinates - sin?
i^0
radicals