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Calculating From The Inside Out

The second principle that will help you calculate a complex expression mentally is to calculate from the ``inside out''. In other words, you calculate in the way that most rapidly reduces the complexity of the expression. Mastering this will help you do very complicated arithmetic almost as easily as with the simple cases.

The basic idea of this is that if you are evaluating some expression numerically - meaning, you have actual numbers to plug in for all the variables and now want to simplify the expression to a single number - you can choose to evaluate parts of the expression in a certain order that maximally simplifies the expression as you go along. At each point, ask yourself, ``What part of this expression can I reduce next that will result in the simplest possible result?'' Often, if you have an expression with many nested expressions, the best path is to calculate the most deeply nested ones first. This results in a simpler expression, which may also have nested elements; you then repeat the process again, only it's easier this time, since the expression has fewer symbols to keep track of.

As an example, consider using the quadratic formula to solve equations of the form $ax^{2}+bx+c=0$. Say you want the solutions of the equation $2.2x^{2}+4x-0.3=0$. Factoring this, mentally or not, would be a lot of work, so it's easiest to just calculate the solution directly with $x=\frac{1}{2a}\left(-b\pm\sqrt{b^{2}-4ac}\right)$.

The rule of thumb to use to decide what part of the expression to reduce next at any given stage is: when deciding which part of an expression to reduce numerically, choose the part of that will result in the expression that has the fewest symbols. In other words, you don't have to evaluate the the expression left-to-right; you choose to evaluate that part of the expression, anywhere within it, that will reduce it the most. When you are reducing a large, complex expression numerically, following this rule makes the process far easier and faster. There are some subtleties to the process, including in combination with windowing, which we will discuss now.

Look at the quadratic formula carefully:

$$x=\frac{1}{2a}\left(-b\pm\sqrt{b^{2}-4ac}\right)$$

Which symbols in the right-hand side can you reduce by plugging in numbers, that will give you an expression that is simpler (in terms of having fewer symbols)? By ``simpler'' here, we basically mean ``easier to work with mentally''. If you reduce the $\frac{1}{2a}$ to (approximately) 0.23, you end up with $0.23\left(-b\pm\sqrt{b^{2}-4ac}\right)$. That's probably a little easier to visualize, but not much. What if you plug in for the -b term? Well, you end up with

$$\frac{1}{2a}\left(-4\pm\sqrt{b^{2}-4ac}\right),$$

which is about the same complexity as the original expression.

The other option is to calculate the discriminant. Depending on your skill and how ``nice'' the numbers are, you may mentally calculate $b^{2}-4ac$ in one step. In this case, that would leave you visualizing $\frac{1}{2a}\left(-b\pm\sqrt{18.64}\right)$. If that is not doable, you can apply the inside-out strategy to the discriminant itself. The inside-out strategy is recursive! Temporarily forget about the larger expression, and just calculate the value of $\sqrt{b^{2}-4ac}$. (This is where windowing comes in.) What part of this expression can you reduce by plugging in numbers, which will simplify the expression the most? The answer is probably the $4ac$ term. Reducing that term gives you the expression $\sqrt{b^{2}+2.64}$. Visualizing that, you can next substitute in for the $b^{2}$ and simplify to get $\sqrt{18.64}$. Since this is not a perfect square, we can choose to approximate a value of 4.4. (The actual value is approximately 4.32.) At this point, you can visualize the original, full expression, which now looks like $\frac{1}{2a}\left(-b\pm4.4\right)$. Much easier to visualize than $\frac{1}{2a}\left(-b\pm\sqrt{b^{2}-4ac}\right)$, isn't it? By properly using the ``inside-out'' strategy, you can rapidly whittle very complex expressions down to more manageable sizes. Since this expression has a plus-or-minus symbol, you will at some point have to split up your image into two parts, one for each solution. Regardless, you can continue reducing it from the inside out in a similar way.