CLEP General Mathematics: Complex Numbers

Instructions:

• Answer 50 questions in 15 minutes.
• If you are not ready to take this test, you can study here.
• 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. Given (4-2i) the complex conjugate would be (4+2i)
Polar Coordinates - sin?
Imaginary number
Real and Imaginary Parts
Complex Conjugate


2. A + bi = z1 c + di = z2 - addition: z1 + z2 = (a + bi) + (c + di) = (a + c) + (b + d)i subtraction: z1 - z2 = (a - c) + (b - d)i
Complex Subtraction
Affix
Complex Numbers: Add & subtract
v(-1)


3. Imaginary number

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4. Have radical
How to add and subtract complex numbers (2-3i)-(4+6i)
non-integers
Complex Number Formula
radicals


5. y / r
zz*
How to multiply complex nubers(2+i)(2i-3)
Polar Coordinates - sin?
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i


6. Every complex number has the 'Standard Form':
has a solution.
x-axis in the complex plane
a + bi for some real a and b.
How to multiply complex nubers(2+i)(2i-3)


7. 2nd. Rule of Complex Arithmetic

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8. The complex number z representing a+bi.
zz*
a + bi for some real a and b.
Affix
Polar Coordinates - z?¹


9. E^(ln r) e^(i?) e^(2pin)
Complex Addition
i^0
Absolute Value of a Complex Number
e^(ln z)


10. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
Integers
multiplying complex numbers
Polar Coordinates - Multiplication
ln z


11. E ^ (z2 ln z1)
a real number: (a + bi)(a - bi) = a² + b²
cos z
imaginary
z1 ^ (z2)


12. All the powers of i can be written as
imaginary
Complex Numbers: Multiply
four different numbers: i - -i - 1 - and -1.
the complex numbers


13. 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
Complex Numbers: Multiply
adding complex numbers
(cos? +isin?)n
Polar Coordinates - Multiplication by i


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

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15. Real and imaginary numbers
complex numbers
standard form of complex numbers
Integers
z - z*


16. 1
the complex numbers
Polar Coordinates - z
Irrational Number
i²


17. Has exactly n roots by the fundamental theorem of algebra
multiply the numerator and the denominator by the complex conjugate of the denominator.
has a solution.
Any polynomial O(xn) - (n > 0)
e^(ln z)


18. R?¹(cos? - isin?)
|z-w|
Polar Coordinates - z?¹
z1 / z2
How to solve (2i+3)/(9-i)


19. I = imaginary unit - i² = -1 or i = v-1
Square Root
How to add and subtract complex numbers (2-3i)-(4+6i)
Imaginary Numbers
Real and Imaginary Parts


20. 2ib
transcendental
z - z*
z1 / z2
Polar Coordinates - Multiplication


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

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22. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Field
imaginary
Any polynomial O(xn) - (n > 0)
Real and Imaginary Parts


23. x / r
Polar Coordinates - cos?
Absolute Value of a Complex Number
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Roots of Unity


24. Not on the numberline
non-integers
De Moivre's Theorem
z1 ^ (z2)
|z| = mod(z)


25. The reals are just the
x-axis in the complex plane
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Complex Number
How to multiply complex nubers(2+i)(2i-3)


26. I
i^1
i²
Polar Coordinates - Division
Complex numbers are points in the plane


27. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Complex Number Formula
Roots of Unity
imaginary
(a + c) + ( b + d)i


28. A complex number and its conjugate
Rules of Complex Arithmetic
conjugate pairs
i^3
i^0


29. A complex number may be taken to the power of another complex number.
How to add and subtract complex numbers (2-3i)-(4+6i)
'i'
Complex Exponentiation
four different numbers: i - -i - 1 - and -1.


30. All numbers
Complex Number
i²
standard form of complex numbers
complex


31. A+bi
subtracting complex numbers
irrational
|z-w|
Complex Number Formula


32. (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)
conjugate
(a + c) + ( b + d)i
v(-1)


33. The square root of -1.
a real number: (a + bi)(a - bi) = a² + b²
Complex Exponentiation
Irrational Number
Imaginary Unit


34. A + bi
z1 / z2
non-integers
Square Root
standard form of complex numbers


35. V(x² + y²) = |z|
Subfield
Polar Coordinates - r
Real and Imaginary Parts
|z-w|


36. Numbers on a numberline
integers
Polar Coordinates - r
Complex Conjugate
multiplying complex numbers


37. 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.
sin iy
Subfield
Complex numbers are points in the plane


38. 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.

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39. 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
-1
Complex Number
How to add and subtract complex numbers (2-3i)-(4+6i)


40. 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
integers
irrational
Any polynomial O(xn) - (n > 0)


41. 2a
a real number: (a + bi)(a - bi) = a² + b²
Field
Square Root
z + z*


42. A number that can be expressed as a fraction p/q where q is not equal to 0.
real
Polar Coordinates - Arg(z*)
a + bi for some real a and b.
Rational Number


43. (e^(-y) - e^(y)) / 2i = i sinh y
z + z*
conjugate pairs
Argand diagram
sin iy


44. ½(e^(-y) +e^(y)) = cosh y
cos iy
can't get out of the complex numbers by adding (or subtracting) or multiplying two
standard form of complex numbers
radicals


45. Equivalent to an Imaginary Unit.
How to multiply complex nubers(2+i)(2i-3)
the distance from z to the origin in the complex plane
Imaginary number
Complex Number


46. Multiply moduli and add arguments
a real number: (a + bi)(a - bi) = a² + b²
multiply the numerator and the denominator by the complex conjugate of the denominator.
Polar Coordinates - Multiplication
z - z*


47. I^2 =
-1
Complex Addition
Liouville's Theorem -
conjugate pairs


48. 3rd. Rule of Complex Arithmetic
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Exponentiation
Real Numbers
i^0


49. 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
non-integers
Rules of Complex Arithmetic
subtracting complex numbers
We say that c+di and c-di are complex conjugates.


50. A subset within a field.
Polar Coordinates - z?¹
complex
Subfield
Absolute Value of a Complex Number