## Archive for the ‘Mathematics’ Category

## Stephen Wolfram: Computing a theory of everything

Posted in Mathematics, Physics on May 6, 2010| Leave a Comment »

## Back to Basics: Negative and Non-Real numbers, an Explanation

Posted in Mathematics, Physics, tagged electron spin, imaginary numbers, math, mathematics, number line, Physics, quantum physics, real and non-real numbers, sign math, two negatives equal a positive on April 29, 2010| 1 Comment »

Recently I’ve been asked two important questions having to do with mathematics that have quite a bit to do with each other. Not only are they linked to each other, but they are linked to understanding how space-time, and all the matter & energy, function. The understanding of basic, fundamental truths of nature, is done through mathematics. It is a language for discussing our physical universe at its most basic level. Understanding the basic concepts of this basic universal language comes easily if represented correctly. It all has to do with why math came about in the first place. Here are the two questions:

**1. Why is it that when you multiply two negative numbers together, the answer is always positive?
**

**2. Why do we need imaginary, or Non-Real numbers?**

The rules, or mechanics of negative number math, such as *“two negative numbers multiplied by each other equal a positive number”* are fine to memorize at first but when imaginary numbers and complex numbers are introduced, inquisitive students of math will want to know why negative math rules exist in the first place.

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**Why is it that when you multiply two negative numbers together, the answer is always positive?**

There are a number of ways to understand this concept, but the way I present here is, I believe, the easiest. Lets begin at the number one. One is most likely the first recognized number in the history of math. We are in fact, one person. If we take that one thing away, we have no things, or nothing represented as* zero*. When adding things to *one* we accumulate and can keep track by counting. We are all familiar with counting and it represents a simple way to use math. As we count, we move away from *zero* in the positive direction. The positive direction can be considered the first, or natural direction of counting. A simple way we can represent this is a person walking forward and counting footsteps.

If this person turns around and walks back to where he came from while counting his steps backwards he is now moving and counting in the opposite of his original direction. The opposite of the original direction is called the negative direction. So when counting forward your moving in the positive direction, and when counting backwards your moving in the negative direction. The fact we always begin counting in the positive direction is important. The act of changing direction while counting in mathematics is indicated by multiplying a number by *-1**.* All this does is change the direction of counting from positive to negative. If you multiply by* *** -1** a second time, it switches back to positive.

Counting can also be represented using a one-dimensional number line. Imagine a point on this number line lying at zero. Imagine this point moving along the number-line in the “*forward*” direction. First it passes one, then two, then three and so on. This is just counting on a number line with a point.

Now imagine the point turning around and going the other way, similar to what the person above did when he reached five footsteps. Remember, this action of turning the other way is indicated in math by multiplying by *-1*. Multiplying by *-1* changes the sign, or direction of the counting. **All a negative sign ever indicates is one switch in direction**. This switch in direction takes place on a one-dimensional line, and there are only two directions to choose from.

Take the numbers 4 and -2 and add them.

4 + (-2) = 4 + (-1)(2) = 2 As you can see, -2 is really just (-1)(2). So, 4 -2 = 4 + -2 = 4 + (-1)(2) = 2 can be read"Count forward four units, turn around count back two units". Lets add two negative numbers:

-5 + (-2) = -1(5)+ -1(2) = -1(5 + 2) = -1(7) = -7

At first it looks like there are two *-1’s,* or “*Turn around”*‘s, but upon further inspection, there is only one. This is because the *-1, *or* *the* *total number of sets is distributed among the sets of five and sets of two. This can be read *“Turn around, count back five units and then two more units”. *The answer will be a negative number.

Multiplication is merely the addition of sets, so essentially, it’s also addition, or counting. We represent five sets of two counting forward like this:

(5)(2) = (2+2+2+2+2) = 10

Since there is no *-1* there is no change in direction and the answer is positive ten.

What about five sets of negative two? It looks like this:

(5)(-2) = (-1)(2)+(-1)(2)+(-1)(2)+(-1)(2)+(-1)(2) = (-1)(2+2+2+2+2) = -10

Remeber that (-1) is distributed among the sets. The commutative property tells us this is the same if the terms switch positions.*“Turn around, count five sets of two”*! .It looks like this:

(-5)(2) = (-1)(5)(2) = (-1)(2+2+2+2+2) = -10

What about if we multiply two negative numbers? This can be said *“Turn around, then turn around again, now count five sets of two” *which looks like this:

(-5)(-2) = (-1)(5)(-1)(2) = (-1)(-1)(5)(2) = (-1)(-1)(2+2+2+2+2) = -(-10) = 10

Notice there are two negative one’s, so we *“turn around” *two times. This brings us back to the forward or positive direction, so our answer is positive!

This can be further simplified by thinking of an arrow on a graph pointing right, or in the *positive x* direction:

The above arrow is pointing in the beginning, or positive direction. In order to point this arrow the opposite way on the *x ax*is we multiply it by *-1*. Now it is an arrow pointing to the left in the *negative x* direction.

Switching directions while counting on a number line is the same as multiplying by *-1*. If we switch directions twice, its simple to see, we are back to moving in the positive direction.

This concept of counting units along one dimension unit lines is an underlaying concept of the very basis of our mathematical language.

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**Why do we need imaginary, or Non-Real numbers? **

Now that we know two negative numbers when multiplied by each other will never be negative, or (-x)(-x)≠ -y This means √(-1) has no REAL answer. In other words, moving forward in our counting, we can say:

(1)(1) = 1, or (-1)(-1) = 1

If we want to do this operation the other way, we say:

√1 = -1 and +1

When we take the square root of a number we are asking what number multiplied by itself equals the number under the radical. The number under the radical, therefore must be positive.

This becomes a problem if we want to solve for a variable in an equation that has an exponent greater than one which happens quite a lot. Lets start with the simplest of this type of equation:

y = x²

Clearly *y *cannot equal a negative number in this equation. In fact, if you graph this equation it forms a parabola that opens in the *positive y* direction. This graph clearly shows that *y* cannot be negative:

If we solve for *x*, then we have:

x = √y where y ≥ 0

Next, lets say we need to solve a series of equations such as the one below:

Find z if z = x² and x = √y and y = -1

The first step in finding *z* is finding *x* which is the square root of *-1*. We cannot say there is No Solution, because the next step is to square √-1. Because we know (√*x*)(√*x*) = *x*, it is intuitive that *z* is going to equal -1, but we cannot complete the calculation without defining what √-1 actually is.

Solving for z, z = x² = (x)(x) = (√y)(√y) = (√-1)(√-1) = .....?...... = -1

What if the equations are much more complicated, but we are still looking for the value of *z*? We need a Non-Real number equal to *√-1* we can use to go from *y* to *z* in the above equations. Sticking with graphs, let’s consider *√1* a unit in the Real world (*1* in the positive or negative direction), and √-1 a unit in the Non-Real world (√-1 = *i*). We can make a graph that uses Non-Real units as the second dimension to our number line we talked about above. Each unit on the Non-Real axis has a value of *i* which equals √-1. This new number line does not replace the *y-axis.* It is an additional non-spacial dimension. This creates a two dimensional graph that represents one space-time dimension and one non-space-time or Non-Real dimension. The graph looks like this:

Imagine our point again on the real number line moving forward, or positive one unit. The line the point traced can be represented as an arrow. Imagine rotating that arrow 180 degrees to backwards. The grey circle represents the path of the spinning arrow.

This movement clearly happens in two dimensions. No matter how small the arrow is, turning around requires at least two dimenstions, but a one dimension line must be able to turn around. It happens all the time whenever we multiply by -1. The arrow would have to turn around instantaneously without passing through a second space dimension or REAL dimension. In other words, a point with direction on a one dimension line changes direction without moving through any other spacial dimensions. This motion can be described using Non-Real numbers and the graph shown above.

With the set of all Non-Real numbers representing the second dimension, we’re able to visualize what happens in the Non-Real second dimension, when a point, or for clarity an arrow, turns from the positive direction to the negative direction.

We just rotated a arrow from positive to negative in one dimension by rotating it through a Non-Real dimension. In the Non-Real realm, the arrow collapses into a spaceless point comprised of enough information to re-built it in the exact opposite direction when it reemerges in the Real dimension. This happens outside of space-time, so no time passes. An instantaneous turn, or switch. With Non-Real numbers, we can look at the information in this Non-Real dimension in the form of complex numbers with understanding they are not Real, but are essential in solving quadratic equations or any algebraic equation with an order higher than one.

Non-real numbers must exist in order for us to have a reference for Real numbers. Similar to the requirement of having backwards if you have forwards. Another way to think of this relationship is to understand the concept that good cannot exist without evil, otherwise, how would we know what evil is? They define each other. This duplicty in nature is everywhere. So, when that point *“turns around”* or* “spins”* to the opposite direction it *“moves”* through the realm of the NON-REAL. It could even be considered that when we count, we count real and non-real units simultaneously, but the non-real units can be thought of as the underlying knowledge that *“forward one”* is also allowing the possibility of *“counting backwards one”*. In other words, with each count of one forward, you create the possibility of counting one backwards. With each real unit forward created, there is a non-real unit forward as well, that is solely knowledge, but can become real if we need to turn around and count the other way. The Non-Real unit allows the change in direction.

This can be further clarified by seeing that our indication for changing direction is really made of Non-Real numbers. Negative one can be factored like this:

-1 = (√-1)² = (i)(i)

So, if we are counting and want to switch directions we multiply by two Non-Real units, or* i²*. Each time we multiply our unit line by *i* it rotates one quadrant in the complex plane.

As you can see, the pattern repeats every forth exponent. For example:

This motion of our unit arrow can now be described by Non-Real numbers. Take 5*i*³. This is equal to 5(*i*)(*i*)(*i*) which can be read: *“Count five, rotate 3 quadrants in the Non-Real space dimension.”* This lands us at -5*i* since *i*³ = –*i*. If this unit arrow rotates one more Non-Real quadrant, it would land on 5 since *i*^4 = 1.

As our arrow spins, its passing through numbers that take the form Real + Non-Real. These are called complex numbers.

The part of the complex number that is Non-Real is non-spacial, so on a one-dimension number-line our arrow with the point at* 0.66+0.75i *would look like an arrow with its tip at 0.66. This can be represented as a projection of our arrow in the complex plane onto the Real axis. As observers in space-time we see this projection:

Since we live in three dimensional space-time we can’t actually see this rotation since it takes place instantaneously, but if we could watch this arrow turn through the complex plane we would only see its projection, or real component on the real axis. To us, it would look like the arrow shrinks until it disappears and then grows out in the negative direction until it gets to negative one, and then start shrinking again. While the arrow passes through the positive Non-Real direction it starts at positive one, shrinks to zero, and extends to negative one. While the arrow passes through the negative Non-Real direction it starts at negative one, shrinks to zero, and extends to positive one. Each real number, thus can be represented by two complex numbers, one in the positive Non-Real direction and one in the negative Non-Real direction. In our example with the arrow, the numbers between* 1* and *-1* are encountered by way of the complex plane by the tip of our arrow twice on each complete rotation. Since this happens instantaneously, a full rotation cannot be perceived, but we can imagine the arrow would be at* 0.66+o.75i* and *0.66-0.75i* at the same time. In fact since the rotation takes no time at all the arrow is in all of its positions in the rotation all at the same time. Since the arrow is in all positions of rotation all at the same time, its reasonable to think of the change in its direction to the negative as more like a change in state than a change in direction.

limited by the dimensions of our perception we cannot observe actions in non-spacial dimensions, but we can look at the information using math. Its important to understand the idea that as we count we observe accumulating units, each observation depends upon the existence of the ability to turn around and count backwards. This ability to turn around in one dimenstion is a property of our space-time and requires a non-space-time dimension or non-real dimension to do so. Each of our three spatial dimensions utilizes some Non-Real dimension. At a quantum level, this property of the universe can be observed. For Example, when an electron is unobserved it only exists as a cloud of probability, but if we observe, (or count) the electron it takes a location in real space within the probability cloud. This location in real space is defined by all the other places in the cloud where it does not exist. As we exist and perceive, we are constantly observing particles, like electrons, and causing them to come into our reality. Our experience, or observations in real space-time are supported by all the other possible experiences that exist only as information in a dimension that has direct interaction with our three spacial and one time dimension. Many speculate this to be a fifth dimension among possibly many more. This topic is much too vast for this simple paper, but having a understanding of these basic math concepts aids in our understanding of the physical world around us. Consider the the following thought.

If movement in our one perceivable time dimension is like movement in our three spacial dimensions, then as we move through time and observe, it can be said we bring what we observe into reality. When un-observed, our future reality exists as information outside of space-time. When observed, not only does it come into our reality, but exists as an expression of everything else that was not brought into our reality at the same time. Having an understanding of imaginary, or Non-Real numbers can help us talk about these ideas and progress humanities overall understanding of reality.

In order to talk about our reality we need a place to begin. Lets begin with the idea of “*one*” and a direction, forward in time. Just like the set of all Non-Real numbers defines and supports the set of all Real numbers in each spacial dimension we can assume the same for the time dimension. Our perception, thus observation of time is supported by all paths of possible events that have NOT taken place and would be considered “non-real” in comparison to our “real” time. Until observed and experienced by us, these paths remain non-real, but must exist outside of space-time as information.