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. All numbers
We say that c+di and c-di are complex conjugates.
ln z
complex
Polar Coordinates - r


2. We see in this way that the distance between two points z and w in the complex plane is
|z| = mod(z)
Integers
z - z*
|z-w|


3. (a + bi) = (c + bi) =
i^2
(a + c) + ( b + d)i
Imaginary number
transcendental


4. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
Complex numbers are points in the plane
Field
i^4
sin z


5. 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 Number Formula
0 if and only if a = b = 0
adding complex numbers
i²


6. The modulus of the complex number z= a + ib now can be interpreted as
Field
the distance from z to the origin in the complex plane
How to find any Power
Every complex number has the 'Standard Form': a + bi for some real a and b.


7. Real and imaginary numbers
complex numbers
cos iy
(a + bi) = (c + bi) = (a + c) + ( b + d)i
The Complex Numbers


8. The complex number z representing a+bi.
standard form of complex numbers
Affix
Every complex number has the 'Standard Form': a + bi for some real a and b.
point of inflection


9. Where the curvature of the graph changes
sin z
Rational Number
point of inflection
v(-1)


10. 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
i^2
Complex Numbers: Add & subtract
Polar Coordinates - Arg(z*)


11. I^2 =
-1
Real and Imaginary Parts
Square Root
Euler Formula


12. Every complex number has the 'Standard Form':
Rules of Complex Arithmetic
a + bi for some real a and b.
integers
sin z


13. 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
De Moivre's Theorem
i^4
Field


14. The square root of -1.
imaginary
Imaginary Unit
(a + c) + ( b + d)i
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i


15. A complex number may be taken to the power of another complex number.
(cos? +isin?)n
Complex Exponentiation
Liouville's Theorem -
interchangeable


16. All the powers of i can be written as
conjugate
four different numbers: i - -i - 1 - and -1.
i^2
Complex numbers are points in the plane


17. When you multiply two complex numbers a + bi and c + di FOIL the terms: (a + bi)(c + di) = (ac - bd) + (ad + bc)i
multiplying complex numbers
radicals
integers
Complex Numbers: Multiply


18. To simplify the square root of a negative number
|z-w|
Polar Coordinates - r
cosh²y - sinh²y
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)


19. V(x² + y²) = |z|
(a + bi) = (c + bi) = (a + c) + ( b + d)i
four different numbers: i - -i - 1 - and -1.
Complex Multiplication
Polar Coordinates - r


20. (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
How to solve (2i+3)/(9-i)
Roots of Unity
Polar Coordinates - Arg(z*)
Polar Coordinates - sin?


21. V(zz*) = v(a² + b²)
|z| = mod(z)
Imaginary Unit
complex
z1 ^ (z2)


22. The product of an imaginary number and its conjugate is
natural
v(-1)
interchangeable
a real number: (a + bi)(a - bi) = a² + b²


23. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Irrational Number
cosh²y - sinh²y
The Complex Numbers
'i'


24. To simplify a complex fraction
Imaginary number
the complex numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.
Euler Formula


25. A number that cannot be expressed as a fraction for any integer.
Irrational Number
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - Division
zz*


26. Any number not rational
irrational
Integers
complex numbers
Complex Numbers: Multiply


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

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28. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
multiply the numerator and the denominator by the complex conjugate of the denominator.
Real and Imaginary Parts
Absolute Value of a Complex Number
ln z


29. For real a and b - a + bi =
zz*
Polar Coordinates - r
adding complex numbers
0 if and only if a = b = 0


30. 2nd. Rule of Complex Arithmetic

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31. R^2 = x
Square Root
Irrational Number
How to solve (2i+3)/(9-i)
Liouville's Theorem -


32. (a + bi)(c + bi) =
Field
i^2 = -1
How to multiply complex nubers(2+i)(2i-3)
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i


33. A complex number and its conjugate
interchangeable
i^3
conjugate pairs
rational


34. 1
cosh²y - sinh²y
Every complex number has the 'Standard Form': a + bi for some real a and b.
Roots of Unity
Subfield


35. Divide moduli and subtract arguments
sin z
Polar Coordinates - Division
cos z
i^1


36. Rotates anticlockwise by p/2
Complex Number Formula
v(-1)
-1
Polar Coordinates - Multiplication by i


37. xpressions such as ``the complex number z'' - and ``the point z'' are now
Subfield
real
interchangeable
Polar Coordinates - r


38. 4th. Rule of Complex Arithmetic
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
(a + bi) = (c + bi) = (a + c) + ( b + d)i
Rules of Complex Arithmetic
complex numbers


39. x / r
|z| = mod(z)
Real Numbers
Polar Coordinates - cos?
Polar Coordinates - z?¹


40. I = imaginary unit - i² = -1 or i = v-1
irrational
Imaginary Numbers
Liouville's Theorem -
i²


41. 5th. Rule of Complex Arithmetic
x-axis in the complex plane
Complex Number
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
i^4


42. Not on the numberline
Real Numbers
How to find any Power
multiplying complex numbers
non-integers


43. 1
v(-1)
How to multiply complex nubers(2+i)(2i-3)
i^2
a + bi for some real a and b.


44. The reals are just the
i^3
multiply the numerator and the denominator by the complex conjugate of the denominator.
e^(ln z)
x-axis in the complex plane


45. 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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46. (e^(iz) - e^(-iz)) / 2i
Complex numbers are points in the plane
standard form of complex numbers
sin z
Affix


47. Imaginary number

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48. A number that can be expressed as a fraction p/q where q is not equal to 0.
Rational Number
Affix
multiply the numerator and the denominator by the complex conjugate of the denominator.
Complex Subtraction


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
Any polynomial O(xn) - (n > 0)
transcendental
We say that c+di and c-di are complex conjugates.
cos iy


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

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