Numbers

Mathematical Constants

Complex Numbers

Some scientific calculators can't handle complex or imaginary numbers. If you happen to have a special calculator that does, probably the manual will tell you how to enter them.

The HP 48 and up series does. It depends on if your calculator is in Polar Coordinate mode or X-Y coordinate mode, but a quick way to get the imaginary number i (regardless of which mode the calculator is currently in), is to press -1, then 'square root' button.

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Numbers

Mathematical Constants

Complex Numbers

First, let's make sure we are not confusing imaginary numbers with complex numbers.

Imaginary (sometimes called "pure imaginary" for clarity) numbers are numbers of the form ai, where a is a real number and i is the principal square root of -1.

To multiply two imaginary numbers ai and bi, start by pretending that i is a variable (like x).

So ai x bi = abi2. But since i is the square root of -1, i2=-1. So abi2=-ab.

For example, 6i x 7i =-42.

5i x 2i =-10.

(-5i) x 2i =-(-10)= 10.

Complex numbers are numbers of the form a+bi, where a and b are real numbers. a is the real part, bi is the imaginary part.

To multiply two complex numbers, again, just treat i as if it were a variable and then in the final answer, substitute -1 wherever you see i2.

Hence (a+bi)(c+di) = ac + adi + bci + dbi2 which simplifies to ac-db + (ad+bc)i.

For example:

(2+3i)(4+5i) = 8 + 10i +12i + 15i2= 8 + 10i + 12i - 15 = -7 + 22i

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Complex Numbers

Possibly because x and y are used to denote the real and imaginary parts, respectively.

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Math and Arithmetic

Algebra

Numbers

Complex Numbers

From the 'origin' move to the left 1 unit, and move down 3 units.

+5i

+4i

+3i

+2i

+1i

-5-4-3-2-1 0 +1+2+3+

-1i

-2i

* -3i

-4i

-5i

The * is at (-1-3i)

whew!

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Computer Terminology

Computer Memory

Complex Numbers

A petabyte, which is 1024 terabytes

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Math and Arithmetic

Algebra

Complex Numbers

Taking the (1/3) power is the same as taking the cube root. The negative power is the same as taking the reciprocal.

So we have x = (-5)-1/3 Cube both sides: x3 = (-5)-1 = -1/5.

There are actually 3 solutions for x. The first one is pretty easy: cube_root(-1/5) which is -cube_root(1/5).

The other two require complex number analysis and are:

- Cube_root(1/5)*(1/2 + i*sqrt(3)/2)
- Cube_root(1/5)*(1/2 - i*sqrt(3)/2)

Where i represents the square root of -1. (Imaginary unit).

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Complex Numbers

An imaginary number is simply a number that contains 'i' which is simply shorthand for the square root of minus one. In the same way, we can write the square root of other negative numbers using i as a factor. For example;

Sqrt(-64) = Sqrt(-1*64) = Sqrt(-1)*Sqrt(64) = i*8 = 8i

There are also 'complex numbers' which are simply combinations of real numbers (all positive and negative numbers) and imaginary numbers. For example;

2+3i

Is a complex number. Although these numbers seem not to "exist" (it's impossible to have a set of 8i golf balls, or for you to weigh 3i kg) they are very useful in the fields of mathematics, physics, and engineering, and allowing their existence can save a lot of trouble when it comes to doing difficult math.

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Complex Numbers

To get the complex conjugate, change the sign in front of the imaginary part. Thus, the complex conjugate of -4 + 5i is -4 - 5i.

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Complex Numbers

To find the complex conjugate of a number, change the sign in front of the imaginary part. Thus, the complex conjugate of 14 + 12i is simply 14 - 12i.

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Math and Arithmetic

Algebra

Complex Numbers

If I understand the question correctly, it is when the algebraic equation (or inequality) is true.

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Safe Driving Techniques

Languages and Cultures

Numbers

Complex Numbers

You will need to ask your driving instructor for it, as the DSA doesn't make that information available to the public.

The ADI number is a unique reference given to every DSA approved driving instructor and can be found on the green badge which all legal instructors should dispay while giving paid instruction.

If you are paying for driving lessons, you have a right to ask to see the instructors licence (trainee instructors have a red/Pink badge), if an instructor can't produce one, they maybe operating illegally.

You don't need an ADI number to book a driving test see ADI number and driving test link below, but you should always make sure that you only take driving lessons from instructors who have a valid licence, check both the photos and expiry date, this will ensure you are getting quality tuition and up to date information regarding the driving test.

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Math and Arithmetic

Algebra

Complex Numbers

8 does not belong

Sequence is to add 1 for the first number, and double the next result, and then add 1 to that result, and double again...

Correct Sequence

2 (+1) 3 (x2) 6 (+1) 7 (x2) 14 (+1) 15 (x2) 30

2-3-6-7-8-14-15-30

(2+1) = (3 x 2) = (6 + 1) = (7 x 2) = (14 + 1) = (15 x 2) = 30 ....

So 8 does not belong in the above series.

282930

Books and Literature

Name Origins

Mathematical Analysis

Complex Numbers

Daphne, a patroness of Diana (Goddess of the Hunt), asked to be turned into a laurel tree after Apollo fell in love with her. She did not love him back and wished to remain an unmarried virgin, true only to Diana, for the rest of her life. Apollo pursued her and she finally begged the gods to be turned into the tree. Thus, to show his devotion, Apollo adopted the laurel tree as his sacred tree and wore the laurel leaves on his head as a crown.

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Algebra

Numbers

Complex Numbers

Mag(3 - 9i) = sqrt(32 + 92) = sqrt(9 + 81) = sqrt(90) or 3*sqrt(10)

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Complex Numbers

complicated numbers

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Numbers

Complex Numbers

A complex number is any number that can be represented as the sum of some number on the real number line and some number on the imaginary number line, a number line perpendicular to the real number line that contains multiples of sqrt(-1), which is more commonly denoted as i.

Complex numbers find applications in AC electronics in calculating impedance. They are also used in producing fractal images, and they turn up in quantum mechanics as well.

The real numbers are a subset of the complex numbers (specifically, they're complex numbers where the imaginary part happens to be zero).

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Business & Finance

Cars & Vehicles

Jobs

Numbers

Complex Numbers

An example: A force of 10 in the x direction and a force of 5 in the y direction will

produce what total force and at what angle? Force = x+iy = 10+5i. The total force will be (10^2+5^2)^.5 or (125)^.5 or ~ 11.18. The angle will be atan y/x or atan 5/10 or .464 or 26.56 degrees.

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Irrational Numbers

Complex Numbers

It would be 8 minus 9i

or 8-9i

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Mathematical Constants

Complex Numbers

Leprecon's only exist after generous portions of Irish stout. How do you know if it's a "he"?

858687

Mathematical Constants

Complex Numbers

1 mol = 6.02x1023 atoms or molecules or individual units

838485

Improving Your Credit Rating

Math and Arithmetic

Prime Numbers

Complex Numbers

A score in time eans 20 years "four score and seven years ago" means 87

313233

Math and Arithmetic

Numbers

Complex Numbers

The root There is some confusion on the questioner's part. A root is a root.

Numbers have many roots:

The square root of 64 is 8 since 8 squared is 64: 8² = 8 × 8 = 64

The cube root of 64 is 4 since 4 cubed is 64: 4³ = 4 × 4 × 4 = 64

The square root of a number x is sometimes called "radical x" because x appear after the radical (or square root) symbol: √x

As square roots are used a lot, it is also often abbreviated from "square root" to just "root", for example √2 can be read as "root 2" though to be strictly correct it is "square root of 2".

Roots also refer to solutions to equations (linear, quadratics, cubics, or higher polynomials) where they equal 0, for example x = -3 and x = 2 are the roots of the equation x² + x - 6 = 0; x = -2, x = 1 and x = 4 are the roots of x³ - 3x² - 6x + 8 = 0.

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Complex Numbers

I found this class that defines complex numbers, and has the capacity of adding them, and much more:

http://www.math.ksu.edu/~bennett/jomacg/c.html

Basically, you define a class with two fields, one for the real part, and one for the imaginary part.

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Complex Numbers

Yes. Traditionally, this line is drawn horizontally, with positive numbers to the right, and negative numbers to the left.

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Numbers

Mathematical Constants

Complex Numbers

One easy way is to use the Quadratic Formula.

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