Imaginary numbers?

Question

Can someone please explain to me how imaginary numbers work, how you square them, etc? A thorough explanation in other words, in simple terms. Please don't direct me to Wikipedia or something. I can't understand it.

Answer 1

i^1 = i
i^2 = -1
i^3 = -i
i^4 = 1
higher powers just cycle through these 4 values.

In the complexe plane, the x-axis is the real axis and the y-axis is the imaginary axiis. The ponit i lives at (0,1). Multiplying by i just rotates the plane counterclockwise by 90 degress.

say (3i)^2 = (3i)*3i = 3*3*i*i = 9 *i^2 = -9

Answer 2

Very, very simply, the imaginary part of an “imaginary” number is the square root of -1, and it is denoted by an i. i x i = -1. For instance the square root of -36 is 6i. And 6i x 6i = -36.

Answer 3

An imaginary number is a real number multiplied by i, which is defined as the square root of -1. It allows you to express the square root of a negative number. If you square any real number, the result will be positive (for example, both 2 and -2 are the square roots of 4). If you square any imaginary number, the result will be negative.

So if you have an imaginary number (anything with i as part of it), ignore the i, square the number normally, and then put a negative sign in front of the result.

Answer 4

An imaginary number is just a representation for the square root of a negative number. This imaginary number can be graphed on a plane or used in later equations, like your question where they can be squared for an actual answer.

The letter i represents SQUARE ROOT -1, which by definition is impossible because you cannot multiply two numbers together and get a negative number.

An equation for a negative number may be represented by something like 3i, which actually represents SQUARE ROOT -9. Regular simplification of roots can be done for example in this equation:

3x^2 + 10 = -26
3x^2 = -36
x^2 = -12
x = (+/-) SQUARE ROOT -12
x = (+/-) i SQUARE ROOT 12
x = (+/-) 2i SQUARE ROOT 3

For any squared number, eg. (3i)^2, the answer will be negative and the coefficient will be squared:

(3i)^2 = -9

in another example:

(2i SQUARE ROOT 5)^2 = -20

Answer 5

Imagine you're standing on a number line, at 0. You are looking in the positive direction, towards "3".

What do you multiply by to do a 180-degree turn, so you're facing towards -3? You multiply by -1, of course:
3 * (-1) = -3
So, multiplying by -1 inverts a quantity. You were facing in the "+" direction, now you're facing in the "-" direction.

But let us say you now want to multiply by some mysterious number (call it x) that corresponds to a 90-degree turn, rather than a 180-degree turn. It is clear that two 90-degree turns make a 180-degree turn. Therefore, if you multiply by x twice (which is the same as multiplying by x^2), it is the same as making two 90-degree turns, which is equivalent to a single 180-degree turn.

This tells us that x^2 = -1. Solving:
x^2 = -1
x = sqrt(-1)
x = i
where i is the imaginary unit.

Thus, the physical interpretation of i is a 90-degree turn. If you multiply by i^2, you have a 180-degree turn. If you multiply by i^4, you have a 360-degree turn and are back where you started. In formal notation:
i^1 = i
i^2 = -1
i^3 = i*i^2 = i*(-1) = -i
i^4 = i^2*i^2 = (-1)*(-1) = 1
i^5 = i
and it wraps back around again.

For example, if you wanted to find the value of i^17, you can simply decrement the exponent by 4 any number of times without changing the value:
i^17 = i^13 = i^9 = i^5 = i^1 = i

Answer 6

i = sqrt(-1)

i^2 = sqrt(-1)sqrt(-1) = - 1

i^3 = i^2(i) = - i

i^4 = (i^2)^2 = (-1)^2 = 1

An imaginary number is usually written as

a + bi

Its conjugate is

a - bi

Their product

(a + bi)(a - bi) - a^2 - abi + abi - b^2i^2 = a^2 + b^2

Answer 7

The imaginary numbers are all multiples of the number i (it's supposed to be italicized.) i is the square root of -1. Since negative numbers cannot have square roots, the imaginary numbers were created. They are drawn perpendicular to a regular number line, to create the complex number plane (looks just like the cartesion coordinate plane, only the y axis is the imaginary numbers). Complex numbers are real number added to imaginary numbers, in the form a + bi, where i is the square root of negative one, b is any real number, and a is any real number. To add complex numbers, just add the reals and the imaginary components separately.
(a + bi) + (c + di) = a + c + (b + d)i
Multiply them using a method called FOIL:
(a + bi)(c + di) = ac + adi + cbi + bdi^2 (i^2 is just -1, so the term becomes -bd)

Whew! I hope that some of that got through to you.

Answer 8

Just concentrate on the fact that i (for imaginary) stands for the square root of -1. It's the solution to x^2 + 1 = 0.

OK, so if i = sqrt(-1), then i^2 = -1.

That's important to really get, so think about it 'til it really sinks in.

Everything else falls in place once you get that i = sqrt(-1) and i^2 = -1.

Then you see, i^3 = i(i^2) = -i, and i^4 = (i^2)(i^2) = 1.

if you're multiply complex numbers, they have an imaginary part. Multiply them out just like F.O.I.L., but keep in mind (always) that i^2 = -1.

That's about it. If you get stuck, just ask.

Have a blessed New Year :)

Answer 9

Ok so an imaginary number is like squaring a negative, its impossible so use imaginary
therefore, the square root of -35 is i times the square root of 35
the square of i is -1