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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
ln z
sin iy
(a + bi) = (c + bi) = (a + c) + ( b + d)i

2. Equivalent to an Imaginary Unit.
'i'
Imaginary number
cosh²y - sinh²y
Euler's Formula

3. Numbers on a numberline
Imaginary number
integers
Roots of Unity
Rational Number

4. We see in this way that the distance between two points z and w in the complex plane is
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Euler's Formula
|z-w|
cos z

5. (e^(-y) - e^(y)) / 2i = i sinh y
sin iy
Complex Conjugate
How to find any Power
How to add and subtract complex numbers (2-3i)-(4+6i)

6. 1
How to find any Power
Polar Coordinates - z?¹
|z| = mod(z)
cosh²y - sinh²y

7. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
Polar Coordinates - Multiplication by i
Complex Subtraction
Complex Exponentiation

8. A number that cannot be expressed as a fraction for any integer.
non-integers
e^(ln z)
Irrational Number
Complex Exponentiation

9. Has exactly n roots by the fundamental theorem of algebra
Polar Coordinates - Division
i^2 = -1
four different numbers: i - -i - 1 - and -1.
Any polynomial O(xn) - (n > 0)

10. x / r
Polar Coordinates - cos?
Absolute Value of a Complex Number
Field
ln z

11. A complex number may be taken to the power of another complex number.
Euler's Formula
i^3
Complex Exponentiation
Polar Coordinates - r

12. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
four different numbers: i - -i - 1 - and -1.
Complex Addition
x-axis in the complex plane

13. A plot of complex numbers as points.
Polar Coordinates - r
Argand diagram
For real a and b - a + bi = 0 if and only if a = b = 0
the complex numbers

14. All the powers of i can be written as
ln z
four different numbers: i - -i - 1 - and -1.
Integers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

15. Where the curvature of the graph changes
point of inflection
four different numbers: i - -i - 1 - and -1.
conjugate
multiply the numerator and the denominator by the complex conjugate of the denominator.

16. A+bi
Polar Coordinates - r
Imaginary Numbers
radicals
Complex Number Formula

17. E ^ (z2 ln z1)
Complex Exponentiation
z1 ^ (z2)
Rules of Complex Arithmetic
the distance from z to the origin in the complex plane

18. x + iy = r(cos? + isin?) = re^(i?)
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - z
Field
Euler Formula

19. To simplify the square root of a negative number
complex numbers
Polar Coordinates - sin?
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Absolute Value of a Complex Number

20. Real and imaginary numbers
complex numbers
-1
point of inflection
Rules of Complex Arithmetic

21. 3rd. Rule of Complex Arithmetic
Complex numbers are points in the plane
Polar Coordinates - Division
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Division

22. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
How to find any Power
the complex numbers
Argand diagram
ln z

23. 2a
cos z
z + z*
transcendental
has a solution.

24. We can also think of the point z= a+ ib as
How to add and subtract complex numbers (2-3i)-(4+6i)
i^4
the vector (a -b)
Subfield

25. Any number not rational
Complex Numbers: Multiply
Polar Coordinates - z?¹
|z| = mod(z)
irrational

26. V(zz*) = v(a² + b²)
Roots of Unity
a real number: (a + bi)(a - bi) = a² + b²
|z| = mod(z)
sin iy

27. z1z2* / |z2|²
z1 / z2
Subfield
transcendental
Complex Number

28. When two complex numbers are divided.
Polar Coordinates - Arg(z*)
Complex Division
Imaginary Unit
'i'

29. ? = -tan?
Complex Numbers: Multiply
Polar Coordinates - Division
Polar Coordinates - Multiplication by i
Polar Coordinates - Arg(z*)

30. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
How to add and subtract complex numbers (2-3i)-(4+6i)
conjugate
subtracting complex numbers
z1 / z2

31. 3
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Rational Number
|z-w|
i^3

32. A² + b² - real and non negative
Polar Coordinates - sin?
Field
Polar Coordinates - z?¹
zz*

33. A complex number and its conjugate
can't get out of the complex numbers by adding (or subtracting) or multiplying two
zz*
conjugate pairs
-1

34. 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
Roots of Unity
Square Root

35. Root negative - has letter i
0 if and only if a = b = 0
i^3
adding complex numbers
imaginary

36. All numbers
Complex Division
Absolute Value of a Complex Number
complex
transcendental

37. R^2 = x
Complex Multiplication
Square Root
ln z
Subfield

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

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39. In this amazing number field every algebraic equation in z with complex coefficients
Euler Formula
Liouville's Theorem -
Rational Number
has a solution.

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.
Complex numbers are points in the plane
Liouville's Theorem -
rational
How to solve (2i+3)/(9-i)

41. 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
z1 ^ (z2)
Irrational Number

42. 2ib
multiply the numerator and the denominator by the complex conjugate of the denominator.
z - z*
0 if and only if a = b = 0
How to solve (2i+3)/(9-i)

43. 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.
Complex Numbers: Multiply
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Division
Affix

44. A + bi
Complex Numbers: Multiply
standard form of complex numbers
sin iy
interchangeable

45. 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
We say that c+di and c-di are complex conjugates.
a real number: (a + bi)(a - bi) = a² + b²
Real and Imaginary Parts
Complex Addition

46. A subset within a field.
the distance from z to the origin in the complex plane
Subfield
z1 / z2
Liouville's Theorem -

47. Have radical
Polar Coordinates - sin?
Complex Exponentiation
radicals
Polar Coordinates - Multiplication

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
rational
subtracting complex numbers
Complex Division
the complex numbers

49. 1
radicals
i^4
zz*
How to find any Power

50. When two complex numbers are subtracted from one another.
Complex Subtraction
0 if and only if a = b = 0
z1 ^ (z2)
subtracting complex numbers