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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)
-1
Complex Conjugate
Field
For real a and b - a + bi = 0 if and only if a = b = 0

2. 1
cosh²y - sinh²y
i^2
'i'
z + z*

3. 4th. Rule of Complex Arithmetic
conjugate pairs
-1
(a + bi) = (c + bi) = (a + c) + ( b + d)i
complex numbers

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

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5. The complex number z representing a+bi.
i^2
Affix
Polar Coordinates - Division
z1 / z2

6. 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
How to solve (2i+3)/(9-i)
e^(ln z)
cosh²y - sinh²y

7. Have radical
four different numbers: i - -i - 1 - and -1.
radicals
Euler Formula
the complex numbers

8. V(x² + y²) = |z|
non-integers
x-axis in the complex plane
Polar Coordinates - r
multiplying complex numbers

9. A number that can be expressed as a fraction p/q where q is not equal to 0.
zz*
Rational Number
(cos? +isin?)n
non-integers

10. Not on the numberline
complex numbers
irrational
non-integers
transcendental

11. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
(cos? +isin?)n
Polar Coordinates - z
The Complex Numbers
Imaginary number

12. We can also think of the point z= a+ ib as
the vector (a -b)
z + z*
Polar Coordinates - r
Polar Coordinates - Multiplication

13. A + bi = z1 c + di = z2 - addition: z1 + z2 = (a + bi) + (c + di) = (a + c) + (b + d)i subtraction: z1 - z2 = (a - c) + (b - d)i
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Every complex number has the 'Standard Form': a + bi for some real a and b.
Imaginary Numbers
Complex Numbers: Add & subtract

14. 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.
cos iy
complex numbers
Integers
Complex Numbers: Multiply

15. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
Any polynomial O(xn) - (n > 0)
'i'
(a + c) + ( b + d)i

16. Numbers on a numberline
Polar Coordinates - z?¹
imaginary
integers
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

17. I = imaginary unit - i² = -1 or i = v-1
z + z*
i^0
Imaginary Numbers
(a + c) + ( b + d)i

18. 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
Real and Imaginary Parts
conjugate
the complex numbers
the distance from z to the origin in the complex plane

19. xpressions such as ``the complex number z'' - and ``the point z'' are now
conjugate pairs
z + z*
interchangeable
a + bi for some real a and b.

20. x / r
i^0
real
Polar Coordinates - cos?
Imaginary Numbers

21. 3rd. Rule of Complex Arithmetic
Complex Number Formula
interchangeable
Field
For real a and b - a + bi = 0 if and only if a = b = 0

22. The field of all rational and irrational numbers.
Every complex number has the 'Standard Form': a + bi for some real a and b.
Real Numbers
Roots of Unity
transcendental

23. The modulus of the complex number z= a + ib now can be interpreted as
a + bi for some real a and b.
the distance from z to the origin in the complex plane
Complex numbers are points in the plane
complex numbers

24. To simplify a complex fraction
0 if and only if a = b = 0
i^3
multiply the numerator and the denominator by the complex conjugate of the denominator.
complex numbers

25. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
Field
can't get out of the complex numbers by adding (or subtracting) or multiplying two
adding complex numbers

26. In this amazing number field every algebraic equation in z with complex coefficients
four different numbers: i - -i - 1 - and -1.
multiplying complex numbers
i^2 = -1
has a solution.

27. Multiply moduli and add arguments
Complex Numbers: Multiply
Polar Coordinates - Multiplication
Imaginary Numbers
non-integers

28. 1
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
i^0
How to multiply complex nubers(2+i)(2i-3)
Affix

29. V(zz*) = v(a² + b²)
Polar Coordinates - cos?
subtracting complex numbers
z + z*
|z| = mod(z)

30. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
Affix
has a solution.
conjugate
Complex numbers are points in the plane

31. All the powers of i can be written as
four different numbers: i - -i - 1 - and -1.
the vector (a -b)
Affix
Euler's Formula

32. Cos n? + i sin n? (for all n integers)
Roots of Unity
Polar Coordinates - z?¹
(cos? +isin?)n
Complex Number

33. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
(a + bi) = (c + bi) = (a + c) + ( b + d)i
0 if and only if a = b = 0
Polar Coordinates - r
Absolute Value of a Complex Number

34. I
Irrational Number
Complex Exponentiation
How to solve (2i+3)/(9-i)
v(-1)

35. The reals are just the
x-axis in the complex plane
interchangeable
z + z*
conjugate

36. When two complex numbers are subtracted from one another.
i^0
Subfield
Complex Subtraction
Polar Coordinates - cos?

37. A subset within a field.
i^0
Complex Subtraction
Subfield
i^1

38. 1st. Rule of Complex Arithmetic
a real number: (a + bi)(a - bi) = a² + b²
Complex Numbers: Add & subtract
i^2 = -1
We say that c+di and c-di are complex conjugates.

39. 3
i^3
irrational
How to find any Power
real

40. Divide moduli and subtract arguments
multiplying complex numbers
the vector (a -b)
i^2 = -1
Polar Coordinates - Division

41. I^2 =
Complex numbers are points in the plane
How to add and subtract complex numbers (2-3i)-(4+6i)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
-1

42. (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
standard form of complex numbers
a real number: (a + bi)(a - bi) = a² + b²
How to multiply complex nubers(2+i)(2i-3)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

43. Root negative - has letter i
imaginary
has a solution.
Imaginary number
Subfield

44. A number that cannot be expressed as a fraction for any integer.
complex
irrational
Irrational Number
How to add and subtract complex numbers (2-3i)-(4+6i)

45. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Real Numbers
Absolute Value of a Complex Number

46. What about dividing one complex number by another? Is the result another complex number? Let's ask the question in another way. If you are given four real numbers a -b -c and d - can you find two other real numbers x and y so that
-1
|z| = mod(z)
Every complex number has the 'Standard Form': a + bi for some real a and b.
We say that c+di and c-di are complex conjugates.

47. To simplify the square root of a negative number
We say that c+di and c-di are complex conjugates.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
radicals
De Moivre's Theorem

48. 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
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
subtracting complex numbers
i^1
How to solve (2i+3)/(9-i)

49. 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
|z| = mod(z)
has a solution.
Real and Imaginary Parts
Complex Number

50. A complex number and its conjugate
conjugate pairs
Complex Numbers: Add & subtract
Complex Multiplication
i^2