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Fusion

Something you will do a lot is take a group of terms and reduce it, replacing the original expression with a smaller one. The simplest example of this is basically arithmetic: taking '2x - x' and replacing it with 'x', for example, or replacing '$1.3+2.2$' with '3.5'. The act of combining symbols like this is called fusion (or symbol fusion, if you want to be verbose.) A fusion is probably the most common kind of morph you'll use. The name is an analogy to nuclear fusion, which happens when two atoms combine to form a single, different atom. When you visualize the equation, you focus on that part of the image that represents the expression you want to change. You visualize the symbols actually moving together, and shifting form into the reduced expression.

As said above, a fusion takes place when you add (or subtract) several terms together to make one term. Another kind of fusion takes place here:

$$\frac{3x+2-2}{x}=\frac{3x}{x}=3$$

This has two fusions, one of which is when the 3x + 2 - 2 is simplified into 3x. The other fusion is a new type, where the x's cancel out in $\frac{3x}{x}$. It may not be obvious that the second one is a fusion. Anytime we take a group of symbols, and replace them with a mathematically equivalent group of fewer symbols, we have fusion. If the expression happens to reduce to zero (when it's being added) or one (when it's being multiplied), so that it would be replaced with nothing, we still call that fusion. This means cancelling out terms in fractions is a kind of fusion.

When you visualize the equation, you can actually visualize the symbols fusing together and morphing into the replacement expression. For an expression like $2y+4-y$, you start by visualizing it,

then use symbol motion to collect the y variable terms,

then chunking them.

Next you imagine the symbols combining:

You then visualize the fused symbols morphing into the replacement:

By the way, when we show an image like this,

it is really shorthand for a sequence like this:

Keep this in mind when you see it later. Imagine the expression collapsing in on itself.

Let's note a few things. First, fusion will not do the math for you. You need to be able to calculate that 2y - y = y. What fusion does is help you focus on changing that particular part of the image (and the equation).

Second, the details of how this is done are somewhat flexible. You don't necessarily have to visualize the process as shown in these figures. The only requirements are that you (a) start with seeing the expression in it starting form, and (b) end with seeing the expression in its reduced (fused) form. So long as you honor those criteria, you can see them moving together and transforming in any kind of sequence or pattern. It may or may not resemble what is shown in this chapter. Maybe you will see the symbols slurping together and reforming in an animation. Or maybe you will not want to have an animation at all; you will instantly substitute the reduced form, or have it go through one or a few intermediate images. Try a few different methods and decide which works best for you.

For the expression $\frac{3x+2-2}{x}$, you start by visualizing it,

then fusing the numerator,

The next step is to cancel out the x. You can envision this as the $\frac{\, x}{x}$ part caving in on itself, morphing into nothing.

See how the chunk has an irregular shape there. You can do that.

Instead of visualizing the fusion happening as shown above, you can also see the parts being struck out, then disappearing:

As is often the case when you are cancelling a term, it is simple enough that you can just do it mentally without any special technique. Thus, you can simply allow the $\frac{\, x}{x}$ drop or fade from the image. The two methods shown here are basically ways of doing that same thing.

Fusion takes place at different levels. The fundamental idea is to take a group of symbols and combine them into a smaller, simpler group. (Of course, they have to be mathematically equal.) The examples of fusion above are basically arithmetic. While this simple level is important, fusion gets more interesting as you use it to make larger 'jumps'. In the example above, if we break it down and write every dinky little step we can think of, it might look like

$$\frac{3x+2-2}{x}=\frac{3x+0}{x}=\frac{3x}{x}=3\times\frac{x}{x}=3\times1=3$$

We skipped most of these steps, and when you process it internally, you likely skip them too. Your level of mastery of math is such that, for example, you can go directly from $3x+2-2$ to 3x. You probably did not stop to think of 3x + 0 explicitly. As an analogy, pretend you are crossing a shallow pond, by stepping on a series of stones. Many of the stones are close together, and you can step on each one if you want. If you have long legs and good balance, you can step on every other stone, or even jump across three or eight at a time, and make it safely across.

You can stretch and pace yourself. Some days, you will be able to take huge jumps - doing several steps at once, except that it will seem like just one step in your mind. The first time you notice this, congratulate yourself, because that is exactly how a talented mathematician does things.

Conversely, if you are tired, you may have trouble taking the kind of jumps you are used to. If that happens, you can just take smaller steps, so to speak. Instead of going from $\frac{3x+2-2}{x}$ directly to 3, take some or all of the intermediate steps above as you work the equation's image.

There is also symbol fission, which is the opposite of symbol fusion. If we replace 2x with 4x - 2x or even 5x - 4x + x, that is fission. You won't split up symbols like this nearly as often, but sometimes doing so is useful, such as when you are factoring polynomials.