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. ? = -tan?
Polar Coordinates - Division
z1 ^ (z2)
Polar Coordinates - Arg(z*)
standard form of complex numbers


2. I
How to multiply complex nubers(2+i)(2i-3)
i^1
cos iy
multiplying complex numbers


3. 1st. Rule of Complex Arithmetic
sin iy
v(-1)
Polar Coordinates - z
i^2 = -1


4. 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
the complex numbers
sin iy
Complex Number
Polar Coordinates - cos?


5. We can also think of the point z= a+ ib as
Polar Coordinates - z?¹
Complex Exponentiation
the vector (a -b)
How to find any Power


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

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7. Multiply moduli and add arguments
has a solution.
-1
Polar Coordinates - Multiplication
sin z


8. E^(ln r) e^(i?) e^(2pin)
sin iy
e^(ln z)
For real a and b - a + bi = 0 if and only if a = b = 0
Complex Multiplication


9. Given (4-2i) the complex conjugate would be (4+2i)
Complex Conjugate
Complex Multiplication
How to solve (2i+3)/(9-i)
the distance from z to the origin in the complex plane


10. A subset within a field.
Subfield
complex numbers
'i'
the vector (a -b)


11. A number that can be expressed as a fraction p/q where q is not equal to 0.
Integers
Euler's Formula
Rational Number
0 if and only if a = b = 0


12. 1
Complex Numbers: Multiply
cos iy
i²
subtracting complex numbers


13. In this amazing number field every algebraic equation in z with complex coefficients
complex numbers
has a solution.
i^1
Imaginary Unit


14. Every complex number has the 'Standard Form':
Subfield
Polar Coordinates - cos?
a + bi for some real a and b.
Polar Coordinates - z


15. To simplify the square root of a negative number
Polar Coordinates - Arg(z*)
Subfield
subtracting complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)


16. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
i^1
conjugate
z1 / z2
Roots of Unity


17. 2nd. Rule of Complex Arithmetic

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18. No i
Imaginary Unit
real
How to solve (2i+3)/(9-i)
imaginary


19. All the powers of i can be written as
Polar Coordinates - Multiplication by i
Any polynomial O(xn) - (n > 0)
For real a and b - a + bi = 0 if and only if a = b = 0
four different numbers: i - -i - 1 - and -1.


20. 1
The Complex Numbers
complex
cosh²y - sinh²y
ln z


21. 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.
the complex numbers
Complex numbers are points in the plane
point of inflection
|z| = mod(z)


22. A complex number and its conjugate
Complex Numbers: Add & subtract
Complex Number
conjugate pairs
Absolute Value of a Complex Number


23. R?¹(cos? - isin?)
Polar Coordinates - z?¹
Rational Number
Polar Coordinates - r
The Complex Numbers


24. (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
integers
cos z
How to solve (2i+3)/(9-i)
subtracting complex numbers


25. z1z2* / |z2|²
Euler Formula
Imaginary Numbers
z1 / z2
(a + bi) = (c + bi) = (a + c) + ( b + d)i


26. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
De Moivre's Theorem
i²
Subfield
ln z


27. The square root of -1.
Square Root
The Complex Numbers
Polar Coordinates - Multiplication
Imaginary Unit


28. Derives z = a+bi
Any polynomial O(xn) - (n > 0)
Complex Number Formula
Euler Formula
Complex Number


29. The complex number z representing a+bi.
Affix
How to multiply complex nubers(2+i)(2i-3)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
De Moivre's Theorem


30. x + iy = r(cos? + isin?) = re^(i?)
Polar Coordinates - z
can't get out of the complex numbers by adding (or subtracting) or multiplying two
interchangeable
Real and Imaginary Parts


31. Not on the numberline
De Moivre's Theorem
non-integers
Affix
Complex Numbers: Multiply


32. Cos n? + i sin n? (for all n integers)
cosh²y - sinh²y
cos iy
(cos? +isin?)n
sin iy


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

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34. We see in this way that the distance between two points z and w in the complex plane is
the complex numbers
Rules of Complex Arithmetic
imaginary
|z-w|


35. I = imaginary unit - i² = -1 or i = v-1
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Imaginary Numbers
z1 / z2
Any polynomial O(xn) - (n > 0)


36. To simplify a complex fraction
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - z?¹
multiply the numerator and the denominator by the complex conjugate of the denominator.
complex numbers


37. The reals are just the
z1 / z2
Imaginary number
How to multiply complex nubers(2+i)(2i-3)
x-axis in the complex plane


38. Root negative - has letter i
imaginary
z + z*
Any polynomial O(xn) - (n > 0)
a + bi for some real a and b.


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
irrational
Real and Imaginary Parts
Polar Coordinates - Multiplication by i
Euler's Formula


40. A plot of complex numbers as points.
Complex Numbers: Multiply
Argand diagram
Complex Multiplication
Irrational Number


41. (e^(iz) - e^(-iz)) / 2i
Complex Numbers: Multiply
sin z
four different numbers: i - -i - 1 - and -1.
imaginary


42. 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
z + z*
Absolute Value of a Complex Number
Complex Number Formula
adding complex numbers


43. 1
i^0
Polar Coordinates - Division
a + bi for some real a and b.
radicals


44. 4th. Rule of Complex Arithmetic
z1 ^ (z2)
(a + bi) = (c + bi) = (a + c) + ( b + d)i
(a + c) + ( b + d)i
For real a and b - a + bi = 0 if and only if a = b = 0


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.'
Rules of Complex Arithmetic
Complex Number
four different numbers: i - -i - 1 - and -1.
-1


46. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
zz*
Field
multiply the numerator and the denominator by the complex conjugate of the denominator.
0 if and only if a = b = 0


47. R^2 = x
Square Root
natural
Complex Number Formula
four different numbers: i - -i - 1 - and -1.


48. For real a and b - a + bi =
a real number: (a + bi)(a - bi) = a² + b²
Polar Coordinates - z
Complex Number
0 if and only if a = b = 0


49. 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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50. Imaginary number

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