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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. Derives z = a+bi
Complex Numbers: Add & subtract
Integers
Complex Number
Euler Formula

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
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Number Formula
a real number: (a + bi)(a - bi) = a² + b²
Real and Imaginary Parts

3. Has exactly n roots by the fundamental theorem of algebra
0 if and only if a = b = 0
How to solve (2i+3)/(9-i)
(cos? +isin?)n
Any polynomial O(xn) - (n > 0)

4. x / r
Polar Coordinates - cos?
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex numbers are points in the plane
i^2 = -1

5. (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
0 if and only if a = b = 0
Polar Coordinates - sin?
How to multiply complex nubers(2+i)(2i-3)
Complex Numbers: Multiply

6. Starts at 1 - does not include 0
Complex Multiplication
natural
(a + bi) = (c + bi) = (a + c) + ( b + d)i
complex numbers

7. V(zz*) = v(a² + b²)
How to multiply complex nubers(2+i)(2i-3)
cos iy
|z| = mod(z)
complex

8. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
conjugate
integers

9. Root negative - has letter i
imaginary
cos z
Euler's Formula
How to add and subtract complex numbers (2-3i)-(4+6i)

10. I
interchangeable
i^1
real
point of inflection

11. A complex number and its conjugate
sin z
Real and Imaginary Parts
conjugate pairs
Complex Subtraction

12. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Absolute Value of a Complex Number
cosh²y - sinh²y
Roots of Unity
Complex Conjugate

13. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
How to multiply complex nubers(2+i)(2i-3)
Absolute Value of a Complex Number
conjugate pairs

14. (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
Field
How to solve (2i+3)/(9-i)
Rational Number
Imaginary number

15. 1
i^4
integers
Real Numbers
Polar Coordinates - Multiplication by i

16. V(x² + y²) = |z|
Polar Coordinates - r
Polar Coordinates - z?¹
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - Arg(z*)

17. I^2 =
The Complex Numbers
-1
Complex Number
Polar Coordinates - cos?

18. The reals are just the
x-axis in the complex plane
Roots of Unity
How to multiply complex nubers(2+i)(2i-3)
0 if and only if a = b = 0

19. 3
Irrational Number
We say that c+di and c-di are complex conjugates.
multiply the numerator and the denominator by the complex conjugate of the denominator.
i^3

20. Like pi
i^4
z - z*
conjugate
transcendental

21. Equivalent to an Imaginary Unit.
subtracting complex numbers
Imaginary number
Complex numbers are points in the plane
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

22. Given (4-2i) the complex conjugate would be (4+2i)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
complex numbers
Complex Conjugate
(a + bi) = (c + bi) = (a + c) + ( b + d)i

23. 2ib
z + z*
z - z*
Liouville's Theorem -
Polar Coordinates - Arg(z*)

24. When two complex numbers are added together.
ln z
Complex Addition
Polar Coordinates - z?¹
e^(ln z)

25. (a + bi) = (c + bi) =
adding complex numbers
Euler Formula
(a + c) + ( b + d)i
i^0

26. The product of an imaginary number and its conjugate is
complex
Imaginary Unit
conjugate pairs
a real number: (a + bi)(a - bi) = a² + b²

27. Divide moduli and subtract arguments
Polar Coordinates - sin?
conjugate
Polar Coordinates - Division
Complex Conjugate

28. (e^(iz) - e^(-iz)) / 2i
sin z
|z-w|
conjugate
Complex Numbers: Add & subtract

29. Have radical
Liouville's Theorem -
the complex numbers
radicals
Complex Number Formula

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

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31. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Square Root
0 if and only if a = b = 0
Integers
adding complex numbers

32. 1
i^3
has a solution.
cosh²y - sinh²y
a real number: (a + bi)(a - bi) = a² + b²

33. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
De Moivre's Theorem
adding complex numbers
sin z
conjugate

34. 2nd. Rule of Complex Arithmetic

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35. R^2 = x
Square Root
i^3
cos iy
Imaginary Unit

36. A subset within a field.
cos iy
Rules of Complex Arithmetic
Complex Addition
Subfield

37. A+bi
Complex Numbers: Multiply
zz*
Complex Number Formula
sin iy

38. When two complex numbers are subtracted from one another.
'i'
i^3
Complex Subtraction
cos iy

39. To simplify the square root of a negative number
has a solution.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Euler's Formula
Any polynomial O(xn) - (n > 0)

40. 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
multiplying complex numbers
Absolute Value of a Complex Number
conjugate
adding complex numbers

41. Where the curvature of the graph changes
point of inflection
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
|z-w|
Complex Conjugate

42. Not on the numberline
non-integers
Polar Coordinates - Multiplication
rational
z1 ^ (z2)

43. 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
Roots of Unity
Complex Multiplication
|z| = mod(z)
the complex numbers

44. ? = -tan?
Subfield
Every complex number has the 'Standard Form': a + bi for some real a and b.
the vector (a -b)
Polar Coordinates - Arg(z*)

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

46. A number that can be expressed as a fraction p/q where q is not equal to 0.
conjugate
Rational Number
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
|z-w|

47. For real a and b - a + bi =
rational
cos iy
Euler Formula
0 if and only if a = b = 0

48. Imaginary number

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49. In this amazing number field every algebraic equation in z with complex coefficients
Polar Coordinates - z?¹
-1
has a solution.
For real a and b - a + bi = 0 if and only if a = b = 0

50. 4th. Rule of Complex Arithmetic
Polar Coordinates - cos?
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - Multiplication by i
conjugate pairs