• Home
• Inner Algebra
• Integrate Your Brain

Windowing

You now have some experience visualizing equations, and working with those images. This is important. As you can tell by now, pretty much everything you learn to do in this book starts with seeing math symbols in your mind's eye.

If you needed to envision the whole equation completely in order to solve it, you could sometimes run into trouble. What if you want to solve an equation that is just too big for you to visualize at this point? Even when you have more practice and become very good at visualizing - or if you are naturally talented at it - bigger and more complex equations will always exist.

Fortunately, you can sidestep all this using a technique called windowing. To use windowing, you do need to be able to remember the equation. The rule of thumb is that you will need to remember it well enough that you could write it down (though you normally would not). At the same time, you would only have to be able to visualize a portion of it at once. Note we are making a distinction between two mental capacities you have: your memory, and your ability to visualize. Most people will find they can exactly remember equations that are much larger than the biggest one they can visualize. Using this ability, you can mentally see enough of the equation to begin processing it.

For equations you can visualize easily enough, you may still want to use windowing, because doing so will let you solve the equation faster and more easily. I actually use windowing for almost every equation I solve mentally, except for very small ones.

What portion of the image (equation) do you need to be able to visualize? It depends. As you have learned, you solve the equations by manipulating the image in particular ways. Normally you are only changing part of the image at any one time. You apply windowing by visualizing those parts of the equation that need to be manipulated, applying the operations to them, and then translating the effect back into your memory of the equation. The key is to become flexible at calling up an image of those parts when you need them. For example, pretend you are solving this equation.

$$\frac{4x+2+\sin2\pi x}{3}=1+2x+\frac{x^{2}}{2}$$

Perhaps you can visualize this whole equation, perhaps not. Let's see how the use of windowing can help us solve this equation without visualizing every bit of it. The next obvious step in solving it is to multiply each side by 3. The way to do this is to visualize the equation, and chunk the 3 on the left side's denominator.

Then, you would move the chunk to the other side like this:

The thing to notice is that only part of the image (the 3 in the left side's denominator) moved. Most of the image did not change at all. Whenever this happens, we have a ripe opportunity to use windowing. See if you can visualize this:

Those are just blurs where the $4x+2+\sin2\pi x$ and $1+2x+\frac{x^{2}}{2}+x^{4}$ used to be. A lot easier to visualize, isn't it? The essence of windowing is to choose to visualize portions of the equation - in this case, the chunked 3 and a few symbols that show the equation's structure - and purposefully allowing the rest of the equation to be foggy. You can visualize those fogged-out parts later, if and when you need them.

You don't have to imagine blurry bars there, by the way - you can visualize squiggly purple lines, empty space, or something else. Experiment, and do what works best for you.

You can move the chunk just like we did a few images ago:

You then ``bring in'' the other symbols as needed. That is, when you are ready to use them, you can take parts of the equation that are foggy and fully visualize them. Since a good next step is to distribute the multiplication on the right hand side, we can bring the symbols that are in the parentheses in next.

Then you can de-chunk the 3 and multiply out the right side like normal.

Alternatively, you can un-blur each of the three terms - 1, 2x, and $\frac{x^{2}}{2}$ - one at a time, multiplying each term by 3 before getting the next.

Windowing is very natural once you get used to it. In fact, you may already have started doing it before reading this chapter, perhaps without realizing it. If you are not at that point yet after working this chapter's exercises, you can get there by introducing it gradually. When you solve equations using symbol motion, visualize the whole equation at first if you need to. Notice if there are parts of the equation you are able to not visualize, even briefly. You can bring those parts back ``in focus'' when they are needed to further solve the equation.