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. The field of all rational and irrational numbers.
Real Numbers
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
e^(ln z)
(cos? +isin?)n


2. 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.'
Polar Coordinates - r
Polar Coordinates - Division
Euler Formula
Complex Number


3. R?¹(cos? - isin?)
cos iy
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
standard form of complex numbers
Polar Coordinates - z?¹


4. When two complex numbers are added together.
Rules of Complex Arithmetic
cos z
Complex Addition
conjugate


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
i²
Rational Number
adding complex numbers
Polar Coordinates - Multiplication by i


6. Ln(r e^(i?)) = ln r + i(? + 2pn) - for all integers n
ln z
Complex numbers are points in the plane
How to solve (2i+3)/(9-i)
Euler's Formula


7. The square root of -1.
Euler Formula
Imaginary Unit
imaginary
Field


8. I
can't get out of the complex numbers by adding (or subtracting) or multiplying two
Complex Subtraction
x-axis in the complex plane
i^1


9. Not on the numberline
Integers
How to multiply complex nubers(2+i)(2i-3)
-1
non-integers


10. 3
i^3
Affix
cos z
z + z*


11. Have radical
Argand diagram
sin z
Complex Number
radicals


12. A plot of complex numbers as points.
Argand diagram
x-axis in the complex plane
Any polynomial O(xn) - (n > 0)
Polar Coordinates - r


13. When two complex numbers are subtracted from one another.
has a solution.
Square Root
adding complex numbers
Complex Subtraction


14. Numbers on a numberline
Real and Imaginary Parts
integers
Polar Coordinates - Multiplication by i
the complex numbers


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

Warning: Invalid argument supplied for foreach() in /var/www/html/basicversity.com/show_quiz.php on line 183


16. The product of an imaginary number and its conjugate is
a real number: (a + bi)(a - bi) = a² + b²
For real a and b - a + bi = 0 if and only if a = b = 0
z - z*
the complex numbers


17. Rotates anticlockwise by p/2
Polar Coordinates - Multiplication by i
zz*
Affix
Field


18. Any number not rational
Subfield
z + z*
complex numbers
irrational


19. The complex number z representing a+bi.
Affix
i²
zz*
Absolute Value of a Complex Number


20. 1
i²
point of inflection
Polar Coordinates - r
Liouville's Theorem -


21. I
Polar Coordinates - z?¹
a real number: (a + bi)(a - bi) = a² + b²
interchangeable
v(-1)


22. The reals are just the
Real Numbers
Square Root
x-axis in the complex plane
radicals


23. Has exactly n roots by the fundamental theorem of algebra
|z| = mod(z)
complex numbers
subtracting complex numbers
Any polynomial O(xn) - (n > 0)


24. A complex number may be taken to the power of another complex number.
De Moivre's Theorem
Complex Exponentiation
Rational Number
Complex Number Formula


25. 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
Any polynomial O(xn) - (n > 0)
a real number: (a + bi)(a - bi) = a² + b²
sin z
the complex numbers


26. Like pi
you write the square root as the product of square roots and simplify: v(-a) = v(-1)v(a) = iv(a)
i^4
Subfield
transcendental


27. When you subtract two complex numbers a + bi and c + di - you get the difference of the real parts and the difference of the imaginary parts: (a + bi) - (c + di) = (a - c) + (b - d)i
subtracting complex numbers
z - z*
Any polynomial O(xn) - (n > 0)
Liouville's Theorem -


28. x / r
Absolute Value of a Complex Number
Every complex number has the 'Standard Form': a + bi for some real a and b.
Polar Coordinates - cos?
Complex numbers are points in the plane


29. Formula: z1 · z2 = (a + bi)(c + di) = ac +adi +cbi +bdi² = (ac - bd) + (ad +cb)i - when you multiply a complex number by its conjugate - you get a real number.
x-axis in the complex plane
Complex Numbers: Multiply
Polar Coordinates - z?¹
Integers


30. A number that cannot be expressed as a fraction for any integer.
Polar Coordinates - r
Every complex number has the 'Standard Form': a + bi for some real a and b.
Irrational Number
Liouville's Theorem -


31. 1
integers
i^2
transcendental
Subfield


32. R^2 = x
point of inflection
i^2 = -1
Complex Numbers: Multiply
Square Root


33. The modulus of the complex number z= a + ib now can be interpreted as
the distance from z to the origin in the complex plane
subtracting complex numbers
a real number: (a + bi)(a - bi) = a² + b²
Complex Division


34. All the powers of i can be written as
a + bi for some real a and b.
four different numbers: i - -i - 1 - and -1.
natural
Polar Coordinates - Multiplication


35. I = imaginary unit - i² = -1 or i = v-1
a + bi for some real a and b.
Real and Imaginary Parts
Imaginary Numbers
Rules of Complex Arithmetic


36. A+bi
z + z*
Complex Number Formula
ln z
Polar Coordinates - z


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

Warning: Invalid argument supplied for foreach() in /var/www/html/basicversity.com/show_quiz.php on line 183


38. One of the numbers ... --2 --1 - 0 - 1 - 2 - ....
Polar Coordinates - Multiplication by i
Integers
four different numbers: i - -i - 1 - and -1.
Imaginary Unit


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.

Warning: Invalid argument supplied for foreach() in /var/www/html/basicversity.com/show_quiz.php on line 183


40. 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
How to add and subtract complex numbers (2-3i)-(4+6i)
i^0
Real and Imaginary Parts
Imaginary Numbers


41. When two complex numbers are multipiled together.
(a + c) + ( b + d)i
Complex Multiplication
Subfield
How to add and subtract complex numbers (2-3i)-(4+6i)


42. (e^(-y) - e^(y)) / 2i = i sinh y
the complex numbers
sin iy
Polar Coordinates - z
subtracting complex numbers


43. 2nd. Rule of Complex Arithmetic

Warning: Invalid argument supplied for foreach() in /var/www/html/basicversity.com/show_quiz.php on line 183


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

Warning: Invalid argument supplied for foreach() in /var/www/html/basicversity.com/show_quiz.php on line 183


45. 1st. Rule of Complex Arithmetic
Absolute Value of a Complex Number
0 if and only if a = b = 0
Complex Number Formula
i^2 = -1


46. To simplify a complex fraction
rational
sin z
complex numbers
multiply the numerator and the denominator by the complex conjugate of the denominator.


47. 5th. Rule of Complex Arithmetic
(a + bi)(c + bi) = ac + bci + adi + bdi^2 =(ac - bc) + (bc + ad)i
v(-1)
Argand diagram
Polar Coordinates - Division


48. A² + b² - real and non negative
zz*
Complex Addition
Real and Imaginary Parts
(cos? +isin?)n


49. (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
Complex Conjugate
How to add and subtract complex numbers (2-3i)-(4+6i)
standard form of complex numbers
How to solve (2i+3)/(9-i)


50. We see in this way that the distance between two points z and w in the complex plane is
Complex Multiplication
Any polynomial O(xn) - (n > 0)
cosh²y - sinh²y
|z-w|