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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. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Rules of Complex Arithmetic
Complex Exponentiation
complex
Integers

2. 1
i^4
Polar Coordinates - Division
For real a and b - a + bi = 0 if and only if a = b = 0
subtracting complex numbers

3. Where the curvature of the graph changes
i^4
subtracting complex numbers
cos z
point of inflection

4. Equivalent to an Imaginary Unit.
the distance from z to the origin in the complex plane
Imaginary number
0 if and only if a = b = 0
Every complex number has the 'Standard Form': a + bi for some real a and b.

5. Real and imaginary numbers
real
ln z
(cos? +isin?)n
complex numbers

6. A + bi
zz*
standard form of complex numbers
Roots of Unity
Polar Coordinates - sin?

7. 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
Euler Formula
Rules of Complex Arithmetic
x-axis in the complex plane
ln z

8. 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
For real a and b - a + bi = 0 if and only if a = b = 0
Polar Coordinates - z
Polar Coordinates - sin?
adding complex numbers

9. Has exactly n roots by the fundamental theorem of algebra
Complex Exponentiation
Any polynomial O(xn) - (n > 0)
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Multiplication

10. 1
i^2
Square Root
e^(ln z)
Integers

11. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
has a solution.
z + z*
e^(ln z)
ln z

12. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
Polar Coordinates - Multiplication
Imaginary Numbers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i

13. 1st. Rule of Complex Arithmetic
i^2 = -1
Subfield
Complex Addition
v(-1)

14. When two complex numbers are divided.
Polar Coordinates - Multiplication
zz*
Complex Division
a + bi for some real a and b.

15. V(x² + y²) = |z|
Polar Coordinates - r
x-axis in the complex plane
Complex Number Formula
Polar Coordinates - Multiplication

16. 2ib
z1 ^ (z2)
Complex Subtraction
z - z*
multiply the numerator and the denominator by the complex conjugate of the denominator.

17. ? = -tan?
Polar Coordinates - Arg(z*)
radicals
Roots of Unity
'i'

18. (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
Argand diagram
Euler's Formula
How to multiply complex nubers(2+i)(2i-3)
interchangeable

19. Root negative - has letter i
For real a and b - a + bi = 0 if and only if a = b = 0
cos iy
Square Root
imaginary

20. The modulus of the complex number z= a + ib now can be interpreted as
the distance from z to the origin in the complex plane
Integers
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
a + bi for some real a and b.

21. 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.'
subtracting complex numbers
Complex numbers are points in the plane
Complex Number
e^(ln z)

22. 1
The Complex Numbers
cosh²y - sinh²y
transcendental
Complex Numbers: Multiply

23. (e^(-y) - e^(y)) / 2i = i sinh y
complex numbers
the complex numbers
Integers
sin iy

24. Starts at 1 - does not include 0
cosh²y - sinh²y
Euler Formula
natural
Subfield

25. A number that cannot be expressed as a fraction for any integer.
Absolute Value of a Complex Number
subtracting complex numbers
Irrational Number
cos z

26. 1
irrational
z1 / z2
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^0

27. To simplify a complex fraction
cosh²y - sinh²y
Real Numbers
Complex Numbers: Multiply
multiply the numerator and the denominator by the complex conjugate of the denominator.

28. ½(e^(-y) +e^(y)) = cosh y
|z-w|
Polar Coordinates - z?¹
cos iy
the distance from z to the origin in the complex plane

29. I
Polar Coordinates - sin?
v(-1)
subtracting complex numbers
Complex Addition

30. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
i^2
Polar Coordinates - z
real

31. z1z2* / |z2|²
point of inflection
z + z*
Polar Coordinates - cos?
z1 / z2

32. 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
z - z*
the complex numbers
Complex Subtraction
has a solution.

33. Written as fractions - terminating + repeating decimals
Polar Coordinates - Multiplication by i
Polar Coordinates - cos?
rational
transcendental

34. xpressions such as ``the complex number z'' - and ``the point z'' are now
interchangeable
a + bi for some real a and b.
i^4
rational

35. 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)
four different numbers: i - -i - 1 - and -1.
ln z
Complex Conjugate

36. Multiply moduli and add arguments
irrational
Rational Number
Complex Number
Polar Coordinates - Multiplication

37. A plot of complex numbers as points.
Complex Subtraction
Rules of Complex Arithmetic
Argand diagram
(a + c) + ( b + d)i

38. (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
Complex Division
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Affix
How to solve (2i+3)/(9-i)

39. 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
Complex Numbers: Multiply
(a + c) + ( b + d)i
subtracting complex numbers

40. No i
|z| = mod(z)
real
the vector (a -b)
can't get out of the complex numbers by adding (or subtracting) or multiplying two

41. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - Division
Complex Multiplication
Any polynomial O(xn) - (n > 0)

42. 4th. Rule of Complex Arithmetic
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Complex Division
De Moivre's Theorem
Imaginary Unit

43. 2nd. Rule of Complex Arithmetic

44. y / r
i^2 = -1
i^0
Polar Coordinates - sin?
Polar Coordinates - z

45. Divide moduli and subtract arguments
(cos? +isin?)n
Square Root
Polar Coordinates - Division
i^0

46. When two complex numbers are added together.
Rules of Complex Arithmetic
Argand diagram
standard form of complex numbers
Complex Addition

47. A² + b² - real and non negative
adding complex numbers
z + z*
zz*
Polar Coordinates - z

48. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
transcendental
Polar Coordinates - Multiplication by i
z1 ^ (z2)

49. The square root of -1.
v(-1)
i^2 = -1
complex
Imaginary Unit

50. 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.