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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. 4th. Rule of Complex Arithmetic
Argand diagram
(a + bi) = (c + bi) = (a + c) + ( b + d)i
How to solve (2i+3)/(9-i)
sin iy

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
Integers
Real and Imaginary Parts
Subfield
conjugate

3. A complex number may be taken to the power of another complex number.
Complex Exponentiation
Rational Number
z - z*
i²

4. (e^(iz) - e^(-iz)) / 2i
Real and Imaginary Parts
Polar Coordinates - Multiplication
multiplying complex numbers
sin z

5. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
adding complex numbers
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - Arg(z*)

6. V(x² + y²) = |z|
zz*
The Complex Numbers
Polar Coordinates - Multiplication by i
Polar Coordinates - r

7. I^2 =
conjugate pairs
multiply the numerator and the denominator by the complex conjugate of the denominator.
-1
multiplying complex numbers

8. Has exactly n roots by the fundamental theorem of algebra
Any polynomial O(xn) - (n > 0)
Polar Coordinates - Arg(z*)
rational
i^0

9. When two complex numbers are subtracted from one another.
Roots of Unity
point of inflection
Complex Subtraction
(a + c) + ( b + d)i

10. When two complex numbers are divided.
i^2 = -1
natural
z - z*
Complex Division

11. Like pi
transcendental
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Polar Coordinates - z
Field

12. Rotates anticlockwise by p/2
For real a and b - a + bi = 0 if and only if a = b = 0
four different numbers: i - -i - 1 - and -1.
How to solve (2i+3)/(9-i)
Polar Coordinates - Multiplication by i

13. x / r
rational
x-axis in the complex plane
Polar Coordinates - cos?
Any polynomial O(xn) - (n > 0)

14. The complex number z representing a+bi.
How to add and subtract complex numbers (2-3i)-(4+6i)
Euler's Formula
Affix
Complex numbers are points in the plane

15. For real a and b - a + bi =
cosh²y - sinh²y
0 if and only if a = b = 0
Polar Coordinates - sin?
Complex Conjugate

16. 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.
a real number: (a + bi)(a - bi) = a² + b²
How to find any Power
Polar Coordinates - sin?
Euler's Formula

17. The product of an imaginary number and its conjugate is
Field
-1
a real number: (a + bi)(a - bi) = a² + b²
cosh²y - sinh²y

18. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
(a + bi) = (c + bi) = (a + c) + ( b + d)i
cos z
imaginary
Integers

19. ½(e^(-y) +e^(y)) = cosh y
Polar Coordinates - Multiplication by i
non-integers
cos iy
Complex Exponentiation

20. A number that cannot be expressed as a fraction for any integer.
Irrational Number
irrational
i^2 = -1
point of inflection

21. To simplify the square root of a negative number
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
multiplying complex numbers
Polar Coordinates - Multiplication
v(-1)

22. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Complex Division
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Field
-1

23. (a + bi) = (c + bi) =
(a + c) + ( b + d)i
natural
i^2
Absolute Value of a Complex Number

24. I
Complex Division
i^1
Every complex number has the 'Standard Form': a + bi for some real a and b.
standard form of complex numbers

25. 1
Complex Multiplication
Integers
i^0
Polar Coordinates - z

26. Written as fractions - terminating + repeating decimals
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - Division
z1 / z2
rational

27. 2nd. Rule of Complex Arithmetic

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28. 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
subtracting complex numbers
sin z
Real Numbers
Imaginary number

29. The square root of -1.
Complex Numbers: Add & subtract
Complex Addition
Imaginary Unit
Complex Multiplication

30. ? = -tan?
Polar Coordinates - Arg(z*)
Euler's Formula
Polar Coordinates - r
a real number: (a + bi)(a - bi) = a² + b²

31. Cos n? + i sin n? (for all n integers)
i^1
De Moivre's Theorem
(cos? +isin?)n
non-integers

32. When two complex numbers are added together.
0 if and only if a = b = 0
subtracting complex numbers
Complex Addition
Complex Number Formula

33. Numbers on a numberline
Polar Coordinates - r
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
subtracting complex numbers
integers

34. 3
conjugate
i^3
Polar Coordinates - Multiplication
Square Root

35. z1z2* / |z2|²
z1 / z2
radicals
Polar Coordinates - sin?
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

36. 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
Imaginary number
standard form of complex numbers
Complex Numbers: Add & subtract
We say that c+di and c-di are complex conjugates.

37. 1
Polar Coordinates - sin?
Irrational Number
i^2
z1 / z2

38. A complex number and its conjugate
Imaginary Numbers
For real a and b - a + bi = 0 if and only if a = b = 0
conjugate pairs
Polar Coordinates - Arg(z*)

39. In this amazing number field every algebraic equation in z with complex coefficients
Imaginary number
How to solve (2i+3)/(9-i)
has a solution.
i^1

40. y / r
Polar Coordinates - sin?
(a + c) + ( b + d)i
Rules of Complex Arithmetic
(cos? +isin?)n

41. 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.
Rules of Complex Arithmetic
Complex Numbers: Multiply
We say that c+di and c-di are complex conjugates.
subtracting complex numbers

42. Given (4-2i) the complex conjugate would be (4+2i)
De Moivre's Theorem
Rules of Complex Arithmetic
Complex Conjugate
conjugate pairs

43. When two complex numbers are multipiled together.
(cos? +isin?)n
Complex Multiplication
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)

44. (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
How to solve (2i+3)/(9-i)
|z| = mod(z)
a real number: (a + bi)(a - bi) = a² + b²
0 if and only if a = b = 0

45. 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 Numbers: Add & subtract
x-axis in the complex plane
the vector (a -b)
Imaginary Unit

46. Imaginary number

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47. 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.'
Complex Number
(cos? +isin?)n
Complex Numbers: Multiply
(a + c) + ( b + d)i

48. Real and imaginary numbers
complex numbers
has a solution.
i²
Polar Coordinates - Division

49. A+bi
sin z
i^0
'i'
Complex Number Formula

50. A subset within a field.
ln z
Affix
Rules of Complex Arithmetic
Subfield