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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. 2nd. Rule of Complex Arithmetic

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2. Multiply moduli and add arguments
Euler Formula
Argand diagram
Polar Coordinates - Multiplication
i^2 = -1

3. 3
e^(ln z)
We say that c+di and c-di are complex conjugates.
i^3
Complex Numbers: Multiply

4. V(x² + y²) = |z|
Complex Number
multiplying complex numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
Polar Coordinates - r

5. To simplify a complex fraction
Polar Coordinates - r
multiply the numerator and the denominator by the complex conjugate of the denominator.
e^(ln z)
i²

6. (e^(iz) - e^(-iz)) / 2i
Complex Division
zz*
z1 / z2
sin z

7. Any set of elements that satisfies the field axioms for both addition and multiplication and is a commutative division algebra.
conjugate pairs
Affix
Field
The Complex Numbers

8. ½(e^(-y) +e^(y)) = cosh y
Rules of Complex Arithmetic
Real and Imaginary Parts
Complex Exponentiation
cos iy

9. y / r
Polar Coordinates - sin?
Rules of Complex Arithmetic
Complex Numbers: Add & subtract
sin iy

10. Has the opposite sign of a complex number; the conjugate of a + bi is a - bi
conjugate
(cos? +isin?)n
Complex Number
Square Root

11. 2ib
conjugate
irrational
z - z*
Imaginary number

12. 2a
a + bi for some real a and b.
z + z*
Complex Number
complex

13. The complex number z representing a+bi.
Imaginary number
Euler's Formula
cosh²y - sinh²y
Affix

14. To simplify the square root of a negative number
i^2 = -1
How to multiply complex nubers(2+i)(2i-3)
Complex Division
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)

15. It is an amazing fact that by adjoining the imaginary unit i to the real numbers we obtain a complete number field called
Complex Multiplication
transcendental
Any polynomial O(xn) - (n > 0)
The Complex Numbers

16. In this amazing number field every algebraic equation in z with complex coefficients
has a solution.
0 if and only if a = b = 0
Rational Number
i^0

17. Solutions to zn = 1 - |z| = 1 - z = e^(i?) - e^(in?) = 1
Roots of Unity
Polar Coordinates - Multiplication
conjugate
Complex Exponentiation

18. x / r
Integers
Polar Coordinates - cos?
transcendental
natural

19. R?¹(cos? - isin?)
Polar Coordinates - z?¹
|z-w|
Complex Subtraction
i^2

20. Equivalent to an Imaginary Unit.
natural
Imaginary number
Any polynomial O(xn) - (n > 0)
i^2

21. Rotates anticlockwise by p/2
conjugate
Complex Conjugate
Polar Coordinates - Multiplication by i
can't get out of the complex numbers by adding (or subtracting) or multiplying two

22. A number that cannot be expressed as a fraction for any integer.
Irrational Number
Argand diagram
standard form of complex numbers
x-axis in the complex plane

23. 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
ln z
the vector (a -b)
Complex Exponentiation
We say that c+di and c-di are complex conjugates.

24. Where the curvature of the graph changes
point of inflection
i^3
zz*
rational

25. z1z2* / |z2|²
Irrational Number
How to multiply complex nubers(2+i)(2i-3)
i^0
z1 / z2

26. A complex number may be taken to the power of another complex number.
i^0
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Complex Exponentiation
(cos? +isin?)n

27. Like pi
the vector (a -b)
Complex Conjugate
transcendental
the complex numbers

28. 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
imaginary
adding complex numbers
standard form of complex numbers
0 if and only if a = b = 0

29. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
Real Numbers
transcendental
Complex Exponentiation

30. 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
radicals
Real and Imaginary Parts
Polar Coordinates - Division
conjugate pairs

31. Have radical
ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
Polar Coordinates - z
conjugate
radicals

32. No i
For real a and b - a + bi = 0 if and only if a = b = 0
Complex numbers are points in the plane
(a + bi) = (c + bi) = (a + c) + ( b + d)i
real

33. A+bi
Complex Division
sin iy
Affix
Complex Number Formula

34. The field of all rational and irrational numbers.
Complex Numbers: Add & subtract
How to solve (2i+3)/(9-i)
Real Numbers
Imaginary Numbers

35. Divide moduli and subtract arguments
Square Root
Polar Coordinates - r
sin iy
Polar Coordinates - Division

36. E^(ln r) e^(i?) e^(2pin)
Polar Coordinates - Multiplication
Argand diagram
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
e^(ln z)

37. Written as fractions - terminating + repeating decimals
Rules of Complex Arithmetic
real
Real and Imaginary Parts
rational

38. The reals are just the
x-axis in the complex plane
cosh²y - sinh²y
Any polynomial O(xn) - (n > 0)
Complex Division

39. 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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40. (2-3i)-(4+6i)you would distribute the negitive and combine your like terms and your answer is -2-9i
Polar Coordinates - z?¹
conjugate
standard form of complex numbers
How to add and subtract complex numbers (2-3i)-(4+6i)

41. A² + b² - real and non negative
zz*
Absolute Value of a Complex Number
i^4
a real number: (a + bi)(a - bi) = a² + b²

42. I
ln z
i^1
can't get out of the complex numbers by adding (or subtracting) or multiplying two
i^4

43. (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
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
How to multiply complex nubers(2+i)(2i-3)
z + z*
Complex Conjugate

44. (e^(-y) - e^(y)) / 2i = i sinh y
Argand diagram
x-axis in the complex plane
sin iy
Polar Coordinates - cos?

45. I^2 =
-1
standard form of complex numbers
'i'
Complex Subtraction

46. Cos n? + i sin n? (for all n integers)
(cos? +isin?)n
Euler Formula
z + z*
z - z*

47. ? = -tan?
Polar Coordinates - Arg(z*)
z1 ^ (z2)
(cos? +isin?)n
point of inflection

48. Root negative - has letter i
complex numbers
Absolute Value of a Complex Number
|z| = mod(z)
imaginary

49. Real and imaginary numbers
Complex Division
Imaginary Unit
z1 ^ (z2)
complex numbers

50. Not on the numberline
Polar Coordinates - Division
interchangeable
non-integers
Affix