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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. To simplify a complex fraction
Imaginary Unit
multiply the numerator and the denominator by the complex conjugate of the denominator.
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
'i'

2. 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.
Every complex number has the 'Standard Form': a + bi for some real a and b.
x-axis in the complex plane
Field
How to find any Power

3. When two complex numbers are multipiled together.
Polar Coordinates - z?¹
conjugate
i^2 = -1
Complex Multiplication

4. (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
How to multiply complex nubers(2+i)(2i-3)
i^1
Polar Coordinates - Division
multiply the numerator and the denominator by the complex conjugate of the denominator.

5. Real and imaginary numbers
complex numbers
e^(ln z)
Roots of Unity
Rational Number

6. The field of all rational and irrational numbers.
'i'
z1 ^ (z2)
Complex Addition
Real Numbers

7. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
Polar Coordinates - Multiplication by i
e^(ln z)
multiplying complex numbers
|z| = mod(z)

8. When two complex numbers are divided.
Complex Division
|z| = mod(z)
(a + c) + ( b + d)i
Subfield

9. 1
Complex Numbers: Multiply
i^2
-1
0 if and only if a = b = 0

10. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
Every complex number has the 'Standard Form': a + bi for some real a and b.
z1 ^ (z2)
sin z

11. Rotates anticlockwise by p/2
(a + bi) = (c + bi) = (a + c) + ( b + d)i
integers
Polar Coordinates - Multiplication by i
Square Root

12. The reals are just the
i^3
Square Root
x-axis in the complex plane
Roots of Unity

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

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14. Not on the numberline
real
z + z*
How to solve (2i+3)/(9-i)
non-integers

15. Have radical
zz*
radicals
multiplying complex numbers
a + bi for some real a and b.

16. A subset within a field.
v(-1)
a real number: (a + bi)(a - bi) = a² + b²
Subfield
i^1

17. (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
Polar Coordinates - r
Complex Subtraction
How to solve (2i+3)/(9-i)
transcendental

18. Divide moduli and subtract arguments
integers
multiplying complex numbers
Polar Coordinates - Division
'i'

19. 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
Every complex number has the 'Standard Form': a + bi for some real a and b.
the complex numbers
We say that c+di and c-di are complex conjugates.
natural

20. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Subfield
Polar Coordinates - r
The Complex Numbers
the distance from z to the origin in the complex plane

21. xpressions such as ``the complex number z'' - and ``the point z'' are now
Rational Number
Imaginary Numbers
interchangeable
Polar Coordinates - Division

22. E^(ln r) e^(i?) e^(2pin)
the vector (a -b)
Any polynomial O(xn) - (n > 0)
Integers
e^(ln z)

23. x / r
Polar Coordinates - cos?
sin z
imaginary
Polar Coordinates - Arg(z*)

24. Starts at 1 - does not include 0
the vector (a -b)
e^(ln z)
i^1
natural

25. No i
Complex Exponentiation
four different numbers: i - -i - 1 - and -1.
De Moivre's Theorem
real

26. Given (4-2i) the complex conjugate would be (4+2i)
Irrational Number
The Complex Numbers
Imaginary Numbers
Complex Conjugate

27. 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
z1 ^ (z2)
Complex Addition
Complex Numbers: Add & subtract

28. 2ib
Irrational Number
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
z - z*
(a + c) + ( b + d)i

29. A number that cannot be expressed as a fraction for any integer.
transcendental
Polar Coordinates - Multiplication
Irrational Number
Liouville's Theorem -

30. All the powers of i can be written as
|z-w|
We say that c+di and c-di are complex conjugates.
non-integers
four different numbers: i - -i - 1 - and -1.

31. A+bi
Complex Number Formula
How to add and subtract complex numbers (2-3i)-(4+6i)
four different numbers: i - -i - 1 - and -1.
0 if and only if a = b = 0

32. A + bi
Polar Coordinates - Multiplication
standard form of complex numbers
Complex Numbers: Add & subtract
For real a and b - a + bi = 0 if and only if a = b = 0

33. The square root of -1.
rational
has a solution.
Imaginary Unit
Polar Coordinates - z?¹

34. 2nd. Rule of Complex Arithmetic

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35. 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.'
Subfield
cosh²y - sinh²y
Complex Number
Complex Subtraction

36. I = imaginary unit - i² = -1 or i = v-1
Rules of Complex Arithmetic
Imaginary Numbers
multiplying complex numbers
Absolute Value of a Complex Number

37. Like pi
transcendental
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Multiplication
point of inflection

38. 3
cos iy
i^3
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
complex

39. To simplify the square root of a negative number
i^2 = -1
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
point of inflection
-1

40. 2a
sin iy
a + bi for some real a and b.
z + z*
Complex Number

41. Cos n? + i sin n? (for all n integers)
has a solution.
Complex numbers are points in the plane
(cos? +isin?)n
the vector (a -b)

42. ½(e^(iz) + e^(-iz))
Rules of Complex Arithmetic
can't get out of the complex numbers by adding (or subtracting) or multiplying two
natural
cos z

43. I
integers
Polar Coordinates - Multiplication
i^1
Imaginary Numbers

44. A² + b² - real and non negative
i^4
zz*
adding complex numbers
transcendental

45. I^2 =
-1
irrational
a real number: (a + bi)(a - bi) = a² + b²
Subfield

46. 5th. Rule of Complex Arithmetic
four different numbers: i - -i - 1 - and -1.
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
How to multiply complex nubers(2+i)(2i-3)
Complex Numbers: Multiply

47. 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.
Affix
Complex numbers are points in the plane
radicals
|z-w|

48. I
Complex Addition
Polar Coordinates - Division
non-integers
v(-1)

49. The product of an imaginary number and its conjugate is
ln z
multiply the numerator and the denominator by the complex conjugate of the denominator.
e^(ln z)
a real number: (a + bi)(a - bi) = a² + b²

50. 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
z1 ^ (z2)
Rational Number
Rules of Complex Arithmetic
can't get out of the complex numbers by adding (or subtracting) or multiplying two