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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. The product of an imaginary number and its conjugate is
multiplying complex numbers
Complex numbers are points in the plane
a real number: (a + bi)(a - bi) = a² + b²
transcendental

2. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
How to add and subtract complex numbers (2-3i)-(4+6i)
Polar Coordinates - sin?
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - cos?

3. 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
Real and Imaginary Parts
Complex Multiplication
subtracting complex numbers
zz*

4. z1z2* / |z2|²
z1 / z2
Euler's Formula
has a solution.
For real a and b - a + bi = 0 if and only if a = b = 0

5. I = imaginary unit - i² = -1 or i = v-1
z + z*
i^2 = -1
transcendental
Imaginary Numbers

6. (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
Polar Coordinates - r
Any polynomial O(xn) - (n > 0)
How to multiply complex nubers(2+i)(2i-3)
conjugate pairs

7. 2a
Real and Imaginary Parts
z + z*
Field
the complex numbers

8. The field of all rational and irrational numbers.
point of inflection
non-integers
Real Numbers
complex

9. Has exactly n roots by the fundamental theorem of algebra
i^2
-1
0 if and only if a = b = 0
Any polynomial O(xn) - (n > 0)

10. ? = -tan?
Rules of Complex Arithmetic
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - Arg(z*)
rational

11. Complex Plane = i - Use the distance formula to determine the point's distance from zero - or - the absolute value.
Liouville's Theorem -
Absolute Value of a Complex Number
rational
can't get out of the complex numbers by adding (or subtracting) or multiplying two

12. R^2 = x
How to add and subtract complex numbers (2-3i)-(4+6i)
Square Root
How to solve (2i+3)/(9-i)
Euler's Formula

13. (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
subtracting complex numbers
radicals
can't get out of the complex numbers by adding (or subtracting) or multiplying two
How to solve (2i+3)/(9-i)

14. Equivalent to an Imaginary Unit.
adding complex numbers
zz*
sin iy
Imaginary number

15. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
four different numbers: i - -i - 1 - and -1.
radicals
a real number: (a + bi)(a - bi) = a² + b²
multiplying complex numbers

16. Like pi
ln z
Complex Numbers: Add & subtract
Polar Coordinates - Arg(z*)
transcendental

17. x + iy = r(cos? + isin?) = re^(i?)
sin z
Liouville's Theorem -
Polar Coordinates - z
Subfield

18. 3
complex numbers
radicals
i^3
z1 / z2

19. V(zz*) = v(a² + b²)
|z| = mod(z)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
real
Polar Coordinates - Division

20. All the powers of i can be written as
Imaginary number
radicals
four different numbers: i - -i - 1 - and -1.
Complex Division

21. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Polar Coordinates - Arg(z*)
Argand diagram
interchangeable
Field

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

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23. Where the curvature of the graph changes
point of inflection
Subfield
Complex Number
Polar Coordinates - Division

24. E ^ (z2 ln z1)
-1
Complex Addition
z1 ^ (z2)
How to solve (2i+3)/(9-i)

25. Have radical
radicals
Absolute Value of a Complex Number
-1
Complex Number Formula

26. y / r
Polar Coordinates - sin?
Imaginary Numbers
(cos? +isin?)n
i^4

27. The square root of -1.
z + z*
Imaginary Unit
Polar Coordinates - Multiplication by i
v(-1)

28. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
z1 / z2
sin iy
conjugate
Argand diagram

29. A² + b² - real and non negative
zz*
|z-w|
has a solution.
i^1

30. ½(e^(iz) + e^(-iz))
cos z
x-axis in the complex plane
v(-1)
0 if and only if a = b = 0

31. All numbers
(cos? +isin?)n
e^(ln z)
Polar Coordinates - z?¹
complex

32. Cos n? + i sin n? (for all n integers)
For real a and b - a + bi = 0 if and only if a = b = 0
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex numbers are points in the plane
(cos? +isin?)n

33. Given (4-2i) the complex conjugate would be (4+2i)
v(-1)
cos z
Complex Conjugate
real

34. 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)
Polar Coordinates - Division
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
v(-1)

35. A number that can be expressed as a fraction p/q where q is not equal to 0.
i²
Rational Number
has a solution.
z1 ^ (z2)

36. (e^(iz) - e^(-iz)) / 2i
Roots of Unity
sin z
the distance from z to the origin in the complex plane
z - z*

37. 1st. Rule of Complex Arithmetic
-1
irrational
i^2 = -1
x-axis in the complex plane

38. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Polar Coordinates - cos?
radicals
Roots of Unity
Affix

39. Not on the numberline
Complex Numbers: Multiply
non-integers
Complex Multiplication
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

40. 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.
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex numbers are points in the plane
Polar Coordinates - Division
Complex Addition

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

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42. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
point of inflection
ln z
multiplying complex numbers
a + bi for some real a and b.

43. 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
(a + bi) = (c + bi) = (a + c) + ( b + d)i
We say that c+di and c-di are complex conjugates.
real

44. In this amazing number field every algebraic equation in z with complex coefficients
cosh²y - sinh²y
Real and Imaginary Parts
has a solution.
Rules of Complex Arithmetic

45. 1
Roots of Unity
Polar Coordinates - sin?
non-integers
cosh²y - sinh²y

46. 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
Euler's Formula
subtracting complex numbers
'i'
multiplying complex numbers

47. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
(a + c) + ( b + d)i
The Complex Numbers
Polar Coordinates - cos?
Complex Numbers: Multiply

48. Any number not rational
z - z*
has a solution.
irrational
multiplying complex numbers

49. When two complex numbers are multipiled together.
We say that c+di and c-di are complex conjugates.
Complex Number
How to find any Power
Complex Multiplication

50. The reals are just the
Subfield
x-axis in the complex plane
i^1
the distance from z to the origin in the complex plane