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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. Where the curvature of the graph changes
point of inflection
We say that c+di and c-di are complex conjugates.
adding complex numbers
interchangeable

2. ½(e^(iz) + e^(-iz))
Polar Coordinates - Arg(z*)
How to find any Power
Polar Coordinates - r
cos z

3. Like pi
Real and Imaginary Parts
non-integers
transcendental
i^4

4. Has exactly n roots by the fundamental theorem of algebra
Complex Numbers: Add & subtract
sin z
Polar Coordinates - Multiplication by i
Any polynomial O(xn) - (n > 0)

5. We see in this way that the distance between two points z and w in the complex plane is
Polar Coordinates - Multiplication
imaginary
|z-w|
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

6. All the powers of i can be written as
ln z
natural
(a + c) + ( b + d)i
four different numbers: i - -i - 1 - and -1.

7. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Argand diagram
Complex Number Formula
Integers
conjugate pairs

8. Root negative - has letter i
Imaginary Numbers
conjugate pairs
imaginary
Complex Addition

9. We consider the a real number x to be the complex number x+ 0i and in this way we can think of the real numbers as a subset of
Any polynomial O(xn) - (n > 0)
real
the complex numbers
Subfield

10. We can also think of the point z= a+ ib as
the vector (a -b)
z + z*
(cos? +isin?)n
z - z*

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
z - z*
i^0
How to multiply complex nubers(2+i)(2i-3)

12. When two complex numbers are multipiled together.
Irrational Number
conjugate
e^(ln z)
Complex Multiplication

13. The modulus of the complex number z= a + ib now can be interpreted as
Complex numbers are points in the plane
the distance from z to the origin in the complex plane
How to multiply complex nubers(2+i)(2i-3)
a + bi for some real a and b.

14. 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
Complex Numbers: Multiply
|z| = mod(z)
Real Numbers

15. A complex number and its conjugate
multiplying complex numbers
conjugate pairs
Complex Subtraction
Complex Addition

16. Not on the numberline
-1
non-integers
Polar Coordinates - Arg(z*)
The Complex Numbers

17. 1
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
transcendental
Complex Division
cosh²y - sinh²y

18. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
De Moivre's Theorem
Any polynomial O(xn) - (n > 0)
-1

19. 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
z + z*
The Complex Numbers
i^3

20. A + bi
complex
standard form of complex numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Liouville's Theorem -

21. I^2 =
transcendental
Rules of Complex Arithmetic
-1
Polar Coordinates - z

22. To simplify the square root of a negative number
real
a real number: (a + bi)(a - bi) = a² + b²
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Multiplication

23. All numbers
complex
Real and Imaginary Parts
ln z
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

24. ? = -tan?
four different numbers: i - -i - 1 - and -1.
Polar Coordinates - Multiplication
Polar Coordinates - Arg(z*)
cosh²y - sinh²y

25. 1. i^2 = -1 2. Every complex number has the 'Standard Form': a + bi for some real a and b. 3. For real a and b - a + bi = 0 if and only if a = b = 0 4. (a + bi) = (c + bi) = (a + c) + ( b + d)i 5. (a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc
Rules of Complex Arithmetic
Polar Coordinates - cos?
Polar Coordinates - Division
complex numbers

26. V(zz*) = v(a² + b²)
How to add and subtract complex numbers (2-3i)-(4+6i)
|z| = mod(z)
Rational Number
Complex Numbers: Multiply

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

28. Multiply moduli and add arguments
Polar Coordinates - Multiplication
sin z
Complex Conjugate
x-axis in the complex plane

29. 1
Polar Coordinates - Arg(z*)
a + bi for some real a and b.
Affix
i^4

30. Numbers on a numberline
Complex Number
ln z
Polar Coordinates - Multiplication by i
integers

31. The product of an imaginary number and its conjugate is
Polar Coordinates - r
Absolute Value of a Complex Number
a real number: (a + bi)(a - bi) = a² + b²
cos iy

32. To simplify a complex fraction
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Division
the complex numbers
Rational Number

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

34. When two complex numbers are added together.
Complex numbers are points in the plane
zz*
Polar Coordinates - sin?
Complex Addition

35. Equivalent to an Imaginary Unit.
(cos? +isin?)n
Imaginary number
the vector (a -b)
has a solution.

36. A+bi
Complex Number
Complex Number Formula
non-integers
interchangeable

37. 1st. Rule of Complex Arithmetic
Complex Numbers: Add & subtract
Absolute Value of a Complex Number
i^2 = -1
Complex Exponentiation

38. Real and imaginary numbers
complex numbers
z + z*
z1 / z2
sin iy

39. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - Arg(z*)
Field
Any polynomial O(xn) - (n > 0)

40. xpressions such as ``the complex number z'' - and ``the point z'' are now
irrational
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
interchangeable
a real number: (a + bi)(a - bi) = a² + b²

41. 2ib
Absolute Value of a Complex Number
z - z*
How to add and subtract complex numbers (2-3i)-(4+6i)
multiplying complex numbers

42. A subset within a field.
Subfield
|z-w|
Affix
irrational

43. (a + bi) = (c + bi) =
Real Numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
real
(a + c) + ( b + d)i

44. 4th. Rule of Complex Arithmetic
Complex Numbers: Add & subtract
(a + bi) = (c + bi) = (a + c) + ( b + d)i
zz*
Polar Coordinates - Arg(z*)

45. 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
-1
Imaginary Numbers
Euler's Formula

46. Derives z = a+bi
i^4
Euler Formula
a real number: (a + bi)(a - bi) = a² + b²
zz*

47. The square root of -1.
0 if and only if a = b = 0
transcendental
Imaginary Unit
Every complex number has the 'Standard Form': a + bi for some real a and b.

48. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
a real number: (a + bi)(a - bi) = a² + b²
z1 / z2
Complex Exponentiation

49. The field of all rational and irrational numbers.
i^0
four different numbers: i - -i - 1 - and -1.
Real Numbers
complex numbers

50. The reals are just the
x-axis in the complex plane
Polar Coordinates - z?¹
Complex Number
subtracting complex numbers