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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. Cos n? + i sin n? (for all n integers)
Polar Coordinates - z?¹
Complex Numbers: Add & subtract
(cos? +isin?)n
i^4

2. R^2 = x
real
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - cos?
Square Root

3. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Liouville's Theorem -
The Complex Numbers
Imaginary Unit
Polar Coordinates - sin?

4. 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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5. E ^ (z2 ln z1)
Complex Division
z1 ^ (z2)
z1 / z2
(cos? +isin?)n

6. All numbers
four different numbers: i - -i - 1 - and -1.
complex
|z-w|
standard form of complex numbers

7. I^26/4= i^24 x i^2 =-1 so u divide the number by 4 and whatevers left over is the number that its equal to.
transcendental
How to find any Power
subtracting complex numbers
Polar Coordinates - Multiplication by i

8. The square root of -1.
natural
Imaginary Unit
Complex numbers are points in the plane
For real a and b - a + bi = 0 if and only if a = b = 0

9. ½(e^(-y) +e^(y)) = cosh y
|z| = mod(z)
Field
four different numbers: i - -i - 1 - and -1.
cos iy

10. 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
Complex Conjugate
De Moivre's Theorem
Polar Coordinates - Multiplication by i
We say that c+di and c-di are complex conjugates.

11. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Absolute Value of a Complex Number
Square Root
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - r

12. 5th. Rule of Complex Arithmetic
four different numbers: i - -i - 1 - and -1.
Imaginary number
conjugate
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

13. V(zz*) = v(a² + b²)
v(-1)
|z| = mod(z)
the distance from z to the origin in the complex plane
Polar Coordinates - Multiplication by i

14. Numbers on a numberline
integers
Roots of Unity
the vector (a -b)
conjugate pairs

15. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
i^1
Polar Coordinates - cos?
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

16. A complex number may be taken to the power of another complex number.
conjugate
Complex Exponentiation
v(-1)
For real a and b - a + bi = 0 if and only if a = b = 0

17. (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
multiply the numerator and the denominator by the complex conjugate of the denominator.
Rational Number
Complex numbers are points in the plane
How to multiply complex nubers(2+i)(2i-3)

18. When two complex numbers are subtracted from one another.
Complex Subtraction
Euler's Formula
Rules of Complex Arithmetic
(a + c) + ( b + d)i

19. Multiply moduli and add arguments
natural
Polar Coordinates - r
Polar Coordinates - Multiplication
sin iy

20. Root negative - has letter i
x-axis in the complex plane
imaginary
How to solve (2i+3)/(9-i)
z1 / z2

21. x / r
(a + c) + ( b + d)i
Complex numbers are points in the plane
Polar Coordinates - cos?
natural

22. The field of numbers of the form - where and are real numbers and i is the imaginary unit equal to the square root of - . When a single letter is used to denote a complex number - it is sometimes called an 'affix.'
i^1
Complex Number
0 if and only if a = b = 0
Polar Coordinates - Multiplication

23. (a + bi) = (c + bi) =
For real a and b - a + bi = 0 if and only if a = b = 0
Integers
(a + c) + ( b + d)i
Polar Coordinates - r

24. Where the curvature of the graph changes
i^2 = -1
sin z
0 if and only if a = b = 0
point of inflection

25. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Roots of Unity
Integers
Polar Coordinates - r
How to find any Power

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

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27. Derives z = a+bi
cos z
Polar Coordinates - Multiplication by i
i^2 = -1
Euler Formula

28. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Polar Coordinates - Division
Polar Coordinates - cos?
How to find any Power
ln z

29. A number that cannot be expressed as a fraction for any integer.
i^2
De Moivre's Theorem
Liouville's Theorem -
Irrational Number

30. 1
cos z
Polar Coordinates - r
How to solve (2i+3)/(9-i)
i^0

31. A subset within a field.
Subfield
Complex Number Formula
'i'
i^0

32. A² + b² - real and non negative
z1 / z2
zz*
v(-1)
Euler's Formula

33. The product of an imaginary number and its conjugate is
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - z?¹
How to find any Power
a real number: (a + bi)(a - bi) = a² + b²

34. (a + bi)(c + bi) =
Roots of Unity
z - z*
i^2 = -1
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

35. 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)
transcendental
Real and Imaginary Parts
adding complex numbers

36. When two complex numbers are added together.
Complex Addition
Complex Numbers: Add & subtract
Imaginary number
e^(ln z)

37. 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
Complex Division
(cos? +isin?)n
Complex Numbers: Add & subtract
(a + c) + ( b + d)i

38. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
Complex Numbers: Add & subtract
conjugate
standard form of complex numbers

39. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
i²
Complex Numbers: Add & subtract
Irrational Number
How to add and subtract complex numbers (2-3i)-(4+6i)

40. Starts at 1 - does not include 0
z1 ^ (z2)
natural
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Numbers: Add & subtract

41. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
has a solution.
Roots of Unity
adding complex numbers

42. I = imaginary unit - i² = -1 or i = v-1
Polar Coordinates - sin?
Imaginary Numbers
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Polar Coordinates - Multiplication

43. (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
(cos? +isin?)n
How to solve (2i+3)/(9-i)
Real and Imaginary Parts
Liouville's Theorem -

44. Written as fractions - terminating + repeating decimals
z + z*
(a + c) + ( b + d)i
rational
Square Root

45. The reals are just the
complex numbers
cos z
x-axis in the complex plane
How to multiply complex nubers(2+i)(2i-3)

46. Has exactly n roots by the fundamental theorem of algebra
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Any polynomial O(xn) - (n > 0)
How to add and subtract complex numbers (2-3i)-(4+6i)
Every complex number has the 'Standard Form': a + bi for some real a and b.

47. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiplying complex numbers
Polar Coordinates - z
z - z*
Square Root

48. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
sin z
Polar Coordinates - sin?
complex

49. Have radical
complex
radicals
Polar Coordinates - cos?
Rules of Complex Arithmetic

50. In this amazing number field every algebraic equation in z with complex coefficients
x-axis in the complex plane
conjugate pairs
has a solution.
i²