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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. To simplify the square root of a negative number
adding complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
We say that c+di and c-di are complex conjugates.
How to solve (2i+3)/(9-i)

2. Divide moduli and subtract arguments
Polar Coordinates - Division
Field
a real number: (a + bi)(a - bi) = a² + b²
v(-1)

3. A + bi
z1 ^ (z2)
complex
standard form of complex numbers
|z-w|

4. Any number not rational
Euler Formula
zz*
irrational
natural

5. (a + bi) = (c + bi) =
complex numbers
standard form of complex numbers
(a + c) + ( b + d)i
|z-w|

6. The square root of -1.
Liouville's Theorem -
Complex Exponentiation
sin z
Imaginary Unit

7. 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
the distance from z to the origin in the complex plane
multiply the numerator and the denominator by the complex conjugate of the denominator.
can't get out of the complex numbers by adding (or subtracting) or multiplying two

8. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Imaginary Unit
ln z
irrational
the distance from z to the origin in the complex plane

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

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10. 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 numbers
Complex Number
ln z
|z| = mod(z)

11. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
imaginary
Integers
can't get out of the complex numbers by adding (or subtracting) or multiplying two
non-integers

12. I
irrational
v(-1)
Roots of Unity
real

13. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - Multiplication by i
How to multiply complex nubers(2+i)(2i-3)
non-integers
Polar Coordinates - z

14. I = imaginary unit - i² = -1 or i = v-1
z1 / z2
i^2 = -1
Imaginary Numbers
z - z*

15. z1z2* / |z2|²
z1 / z2
Real and Imaginary Parts
i^2 = -1
can't get out of the complex numbers by adding (or subtracting) or multiplying two

16. The complex number z representing a+bi.
(cos? +isin?)n
Affix
Euler Formula
(a + c) + ( b + d)i

17. x / r
How to find any Power
Polar Coordinates - cos?
Field
For real a and b - a + bi = 0 if and only if a = b = 0

18. A subset within a field.
Polar Coordinates - r
Subfield
multiply the numerator and the denominator by the complex conjugate of the denominator.
e^(ln z)

19. 1
sin z
i^4
rational
Complex Conjugate

20. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
standard form of complex numbers
The Complex Numbers
x-axis in the complex plane
multiplying complex numbers

21. 1
i²
Absolute Value of a Complex Number
complex
a + bi for some real a and b.

22. Like pi
conjugate pairs
Euler's Formula
Absolute Value of a Complex Number
transcendental

23. Root negative - has letter i
Any polynomial O(xn) - (n > 0)
Integers
real
imaginary

24. When two complex numbers are subtracted from one another.
sin z
interchangeable
Complex Subtraction
Polar Coordinates - Multiplication

25. A² + b² - real and non negative
Every complex number has the 'Standard Form': a + bi for some real a and b.
Real and Imaginary Parts
zz*
-1

26. When two complex numbers are multipiled together.
Complex Multiplication
Polar Coordinates - Division
How to multiply complex nubers(2+i)(2i-3)
standard form of complex numbers

27. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
(cos? +isin?)n
transcendental
Roots of Unity
v(-1)

28. 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 Number Formula
cos iy
Complex Addition
Complex Numbers: Add & subtract

29. Equivalent to an Imaginary Unit.
standard form of complex numbers
the distance from z to the origin in the complex plane
Imaginary number
Rules of Complex Arithmetic

30. I
i^1
sin z
Affix
irrational

31. 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.
How to find any Power
For real a and b - a + bi = 0 if and only if a = b = 0
cos z
point of inflection

32. (e^(-y) - e^(y)) / 2i = i sinh y
Complex Number Formula
sin iy
Polar Coordinates - z
irrational

33. 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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34. When two complex numbers are added together.
Rational Number
Complex Addition
the distance from z to the origin in the complex plane
subtracting complex numbers

35. 1
cosh²y - sinh²y
adding complex numbers
Polar Coordinates - z?¹
Integers

36. Real and imaginary numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
complex numbers
Square Root
Imaginary Unit

37. A complex number may be taken to the power of another complex number.
Imaginary Numbers
Complex Exponentiation
How to solve (2i+3)/(9-i)
Irrational Number

38. Numbers on a numberline
subtracting complex numbers
De Moivre's Theorem
Imaginary Unit
integers

39. A complex number and its conjugate
conjugate pairs
x-axis in the complex plane
four different numbers: i - -i - 1 - and -1.
Imaginary Numbers

40. 2ib
z - z*
Imaginary Numbers
Polar Coordinates - z
Every complex number has the 'Standard Form': a + bi for some real a and b.

41. A number that can be expressed as a fraction p/q where q is not equal to 0.
Subfield
Rational Number
i^0
integers

42. 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
Complex Multiplication
Real and Imaginary Parts
De Moivre's Theorem
Polar Coordinates - z?¹

43. ½(e^(-y) +e^(y)) = cosh y
cos iy
Every complex number has the 'Standard Form': a + bi for some real a and b.
z + z*
Complex Number

44. In the same way that we think of real numbers as being points on a line - it is natural to identify a complex number z=a+ib with the point (a -b) in the cartesian plane.
Complex Exponentiation
Complex numbers are points in the plane
Complex Addition
How to add and subtract complex numbers (2-3i)-(4+6i)

45. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
imaginary
natural
Field
Imaginary Unit

46. 2nd. Rule of Complex Arithmetic

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47. A number that cannot be expressed as a fraction for any integer.
Irrational Number
Absolute Value of a Complex Number
De Moivre's Theorem
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

48. R?¹(cos? - isin?)
Polar Coordinates - z?¹
complex
Polar Coordinates - sin?
Every complex number has the 'Standard Form': a + bi for some real a and b.

49. No i
Argand diagram
i^0
real
a real number: (a + bi)(a - bi) = a² + b²

50. (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
conjugate
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to multiply complex nubers(2+i)(2i-3)
rational