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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. 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
natural
Polar Coordinates - sin?
irrational
Real and Imaginary Parts

2. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
a + bi for some real a and b.
sin z
standard form of complex numbers

3. A number that can be expressed as a fraction p/q where q is not equal to 0.
Imaginary number
Rational Number
-1
complex numbers

4. Like pi
Absolute Value of a Complex Number
transcendental
complex numbers
ln z

5. When two complex numbers are divided.
Complex Division
i^2
e^(ln z)
i^3

6. R^2 = x
Real and Imaginary Parts
Field
transcendental
Square Root

7. When two complex numbers are subtracted from one another.
Complex Subtraction
(a + c) + ( b + d)i
Complex numbers are points in the plane
|z-w|

8. Multiply moduli and add arguments
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - Multiplication
the complex numbers
Complex numbers are points in the plane

9. Written as fractions - terminating + repeating decimals
Complex Numbers: Add & subtract
i^3
rational
the complex numbers

10. I
Integers
i^1
cos iy
Imaginary number

11. 1
standard form of complex numbers
i²
De Moivre's Theorem
Roots of Unity

12. Have radical
How to solve (2i+3)/(9-i)
z1 / z2
How to find any Power
radicals

13. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Euler's Formula
Polar Coordinates - Arg(z*)
has a solution.
Absolute Value of a Complex Number

14. 1
rational
i^4
-1
adding complex numbers

15. (a + bi)(c + bi) =
i²
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
e^(ln z)
Irrational Number

16. We can also think of the point z= a+ ib as
Complex Numbers: Add & subtract
the vector (a -b)
cos iy
i^4

17. When two complex numbers are added together.
Field
e^(ln z)
x-axis in the complex plane
Complex Addition

18. The product of an imaginary number and its conjugate is
-1
(a + c) + ( b + d)i
a real number: (a + bi)(a - bi) = a² + b²
Complex Exponentiation

19. The complex number z representing a+bi.
Affix
complex numbers
z + z*
cos z

20. (e^(iz) - e^(-iz)) / 2i
x-axis in the complex plane
sin z
Complex Division
non-integers

21. Not on the numberline
natural
non-integers
Square Root
irrational

22. ½(e^(iz) + e^(-iz))
Complex numbers are points in the plane
cos z
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Absolute Value of a Complex Number

23. 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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24. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
Polar Coordinates - z
Euler Formula
|z| = mod(z)

25. A complex number may be taken to the power of another complex number.
has a solution.
z1 ^ (z2)
e^(ln z)
Complex Exponentiation

26. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Imaginary Numbers
How to add and subtract complex numbers (2-3i)-(4+6i)
has a solution.
How to solve (2i+3)/(9-i)

27. Derives z = a+bi
Euler Formula
Liouville's Theorem -
rational
Complex Subtraction

28. A complex number and its conjugate
i^0
Polar Coordinates - sin?
conjugate pairs
0 if and only if a = b = 0

29. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Integers
z1 / z2
conjugate pairs

30. 3rd. Rule of Complex Arithmetic
Polar Coordinates - Multiplication by i
For real a and b - a + bi = 0 if and only if a = b = 0
conjugate pairs
How to find any Power

31. I = imaginary unit - i² = -1 or i = v-1
Polar Coordinates - Arg(z*)
Polar Coordinates - r
Imaginary Numbers
non-integers

32. Numbers on a numberline
Polar Coordinates - Multiplication
Imaginary number
Imaginary Unit
integers

33. E ^ (z2 ln z1)
Polar Coordinates - sin?
z1 ^ (z2)
i²
De Moivre's Theorem

34. 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
How to add and subtract complex numbers (2-3i)-(4+6i)
How to multiply complex nubers(2+i)(2i-3)

35. The reals are just the
Imaginary Unit
Complex Numbers: Add & subtract
x-axis in the complex plane
Polar Coordinates - r

36. 3
i^3
Polar Coordinates - Multiplication by i
sin iy
i²

37. A+bi
The Complex Numbers
sin iy
Complex Number Formula
Complex Multiplication

38. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Polar Coordinates - r
Field
i^1
The Complex Numbers

39. ? = -tan?
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - Arg(z*)
Imaginary Numbers
Complex Exponentiation

40. 4th. Rule of Complex Arithmetic
De Moivre's Theorem
(cos? +isin?)n
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Imaginary number

41. Rotates anticlockwise by p/2
cos z
Polar Coordinates - Arg(z*)
-1
Polar Coordinates - Multiplication by i

42. A + bi
Rational Number
Euler's Formula
Complex Exponentiation
standard form of complex numbers

43. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
How to add and subtract complex numbers (2-3i)-(4+6i)
'i'
sin iy
ln z

44. 2a
z + z*
a + bi for some real a and b.
The Complex Numbers
Real Numbers

45. (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
Polar Coordinates - sin?
Complex numbers are points in the plane
How to find any Power
How to solve (2i+3)/(9-i)

46. ½(e^(-y) +e^(y)) = cosh y
cos iy
Imaginary number
De Moivre's Theorem
0 if and only if a = b = 0

47. We see in this way that the distance between two points z and w in the complex plane is
|z-w|
Complex Division
Imaginary Numbers
the vector (a -b)

48. A² + b² - real and non negative
zz*
|z| = mod(z)
Imaginary Numbers
Complex Number

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

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50. A number that cannot be expressed as a fraction for any integer.
Complex Numbers: Add & subtract
Imaginary number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Irrational Number