What Is 2i Equal To

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Multiplying complex numbers

Multiplication washed algebraically. Circuitous multiplication is a more difficult operation to understand from either an algebraic or a geometric betoken of view. Let's do it algebraically kickoff, and let'south take specific circuitous numbers to multiply, say 3 + 2i and 1 + 4i. Each has 2 terms, so when we multiply them, nosotros'll get iv terms:

(three + 2i)(1 + ivi) = 3 + 12i + 2i + eighti ².

Now the 12i + 2i simplifies to fourteeni, of course. What virtually the 8i ⁱⁱ? Remember we introduced i equally an abridgement for √–1, the foursquare root of –1. In other words, i is something whose foursquare is –one. Thus, 8i ² equals –eight. Therefore, the product (three + twoi)(one + ivi) equals –5 + 14i.

If y'all generalize this instance, y'all'll get the general rule for multiplication

(x +yi)(u +vi) = (xu –yv) + (fifteen +yu)i.

Remember that (xu –yv), the real part of the production, is the product of the existent parts minus the product of the imaginary parts, simply (xv +yu), the imaginary part of the product, is the sum of the two products of one real part and the other imaginary part.

Let's look at some special cases of multiplication.

Multiplying a complex number by a real number. In the above formula for multiplication, if 5 is zero, then you go a formula for multiplying a circuitous number x +yi and a real number u together:

(x +yi)u = xu +yu i.

In other words, you just multiply both parts of the complex number by the real number. For case, 2 times 3 +i is just half-dozen + iii. Geometrically, when you double a complex number, just double the altitude from the origin, 0. Similarly, when you lot multiply a complex number z past ane/2, the result will exist half way between 0 and z. Yous can recollect of multiplication by two as a transformation which stretches the complex plane C past a cistron of two abroad from 0; and multiplication by 1/two as a transformation which squeezes C toward 0.

Multiplication and accented value. Even though nosotros've only washed i instance for multiplication, it'southward plenty to suggest that the absolute value of zw (i.due east., altitude from 0 to zw) might be the absolute value of z times the absolute value of w. It was when west was the real number u just above. In fact, this is true in general:

|zw| = |z| |w|

The verification of this identity is an practice in algebra. In order to prove information technology, we'll prove it's true for the squares so we don't take to bargain with square roots. We'll evidence |zw|² = |z|²|due west|². Let z be ten +yi, and let w exist u +vi. Then, according to the formula for multiplication, zw equals (xu –yv) + (xv +yu)i. Recall from the section on accented values that

|z|^two = x ⁱⁱ + y ²

Similarly, we take

|west|² = u ⁱⁱ + v ²

and, since zw = (xu –yv) + (xv +yu)i,

|wz|ⁱⁱ = (xu –yv)ⁱⁱ + (xv +yu)ⁱⁱ

And then, in order to show |zw|² = |z|²|due west|^two, all you lot have to do is prove that

(xu –yv)² + (xv +yu)² = (x ⁱⁱ + y ²) (u ² + five ²)

and that's a straightforward exercixe in algebra.

Powers of i. For our next special instance of multiplication, consider the various powers of the imaginary unit i. We started with the supposition that i ² = –1. What nearly i ^three? It's but i ⁱⁱ times i, and that's –1 times i. Therefore, i ⁱⁱⁱ = –i. That'south interesting: the cube of i is its ain negation. Adjacent consider i ⁴. That's the square of i ², that is, the square of –1. So i ⁴ = 1. In other words, i is a fourth root of 1. You can show that –i is some other 4th root of i. And since both –ane and ane are foursquare roots of ane, nosotros at present know all iv fourth roots of one, namely, 1, i, –1, and –i. This observation connects to the Fundamental Theorem of Algebra since the equation z ⁴ = 1 is a fourth-degree equation and so must take exactly 4 roots.

College powers of i are piece of cake to find now that nosotros know i ⁴ = ane. For example, i ⁵ is i times i ⁴, and that's just i. You can reduce the power of i by four and non change the result. For another example, i ¹¹ = i ^vii = i ^three = –i.

How most negative powers of i? What is the reciprocal of i, that is, i ^–1? For the same reason that you can subtract 4 from a power of i and not alter the result, y'all can also add four to the power of i. That means i ^–one = i ³ = –i. Thus, the reciprocal of i is –i. Imagine–a number whose reciprocal is its ain negation! Of course, information technology'southward easy to cheque that i times –i is 1, so, of course, i and –i are reciprocals.

Roots of unity. The diverse roots of 1 are called roots of unity. In general, by the Fundamental Theorem of Algebra, the number of n-th roots of unity is n, since at that place are n roots of the due north-thursday caste equation z ^u – 1 = 0. The square roots of unity are one and –1. The fourth roots are ±i, ±i, as noted before in the section on absolute value. Also, in that section, it was mentioned that ±√2/two ±i√2/2 were square roots of i and –i, and now with the formula for multiplication, that'south easy to verify. Therefore, the eight eight-roots of unity are ±i, ±i, and ±√2/2 ±i√two/2. Notice how these eight roots of unity are equally spaced around the unit circle.

We can use geometry to find some other roots of unity, in particular the cube roots and sixth roots of unity. Just let'south wait a little bit for them.

Multiplying a complex number by i. In our goal toward finding a geometric interpretation of complex multiplication, let'south consider next multiplying an capricious complex number z =10 +yi by i.

zi = (x +yi)i = –y +xi.

Let'south interpret this statement geometrically. The point z in C is located x units to the right of the imaginary centrality and y units above the real axis. The point zi is located y units to the left, and x units to a higher place. What has happened is that multiplying by i has rotated to betoken z 90° counterclockwise effectually the origin to the point zi. Stated more briefly, multiplication past i gives a ninety° counterclockwise rotation about 0.

You tin analyze what multiplication past –i does in the aforementioned mode. Yous'll notice that multiplication by –i gives a ninety° clockwise rotation about 0. When nosotros don't specify counterclockwise or clockwise when referring to rotations or angles, nosotros'll follow the standard convention that counterclockwise is intended. And so nosotros can say that multiplication past –i gives a –90° rotation about 0, or if you prefer, a 270° rotation virtually 0.

A geometric interpretation of multiplication. To completely justify what we're near to see, trigonometry is needed, and that is washed in an optional section. For now, we'll run into the results without the justification. We've seen 2 special cases of multiplication, i past reals which leads to scaling, the other by i which leads to rotation. The full general example is a combination of scaling and rotation.

Allow z and w be points in the complex plane C. Draw the lines from 0 to z, and 0 to w. The lengths of these lines are the accented values |z| and |w|, respectively. Nosotros already know the length of the line from 0 to zw is going to be the accented value |zw| which equals |z| |w|. (In the diagram, |z| is about 1.6, and |w| is about 2.ane, so |zw| should exist most 3.four. Note that the unit circle is shaded in.) What we don't know is the direction of the line from 0 to zw.

The answer is that "angles add". Nosotros'll determine the direction of the line from 0 to z past a sure angle, called the argument of z, sometimes denoted arg(z). This is the angle whose vertex is 0, the first side is the positive real axis, and the second side is the line from 0 to z. The other point w angle arg(w). Then the product zw volition accept an bending which is the sum of the angles arg(z) + arg(due west). (In the diagram, arg(z) is about 20°, and arg(w) is about 45°, so arg(zw) should be about 65°.)

In summary, we take ii equations which decide where zw is located in C:

|zw| = |z| |westward|

arg(zw) = arg(z) + arg(w)

Next section: Angles and polar coordinates

Previous section: Absolute value

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Electronic mail: djoyce@clarku.edu

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