5.2: Fractional Changes and Low-Entropy Expressions

Using the one-or-few method for mental multiplication is fast. For example, \(3.15 \times 7.21\) quickly becomes \(\text{few} \times 101 ∼ 30\), which is within 50% of the exact product 22.7115. To get a more accurate estimate, round 3.15 to 3 and 7.21 to 7. Their product 21 is in error by only 8%. To reduce the error further, one could split \(3.15 × 7.21\) into a big part and an additive correction. This decomposition produces

\[\begin{align*} 3.15 \times 7.21 &= (3 + 0.15)(7 + 0.21) \\[4pt] &= \underbrace{3 \times 7}_{\text{big part}} + \underbrace{0.15 \times 7 + 3 \times 0.21 + 0.15 \times 0.21}_{\text{additive correction}}. \end{align*} \nonumber \]

The approach is sound, but the literal application of taking out the big part produces a messy correction that is hard to remember and understand. Slightly modified, however, taking out the big part provides a clean and intuitive correction. As gravy, developing the improved correction introduces two important street-fighting ideas: fractional changes (Section 5.2.1) and low-entropy expressions (Section 5.2.2). The improved correction will then, as a first of many uses, help us estimate the energy saved by highway speed limits (Section 5.2.3).

Fractional Changes

The hygienic alternative to an additive correction is to split the product into a big part and a multiplicative correction:

\[3.15 \times 7.21 = \underbrace{3 \times 7}_{\text{big part}} \times \underbrace{(1 + 0.05) \times (1 + 0.03)}_{\text{correction factor}}. \label{5.4} \]

The correction factor is the area of a rectangle with width \(1 + 0.05\) and height \(1 + 0.03\).

The rectangle contains one sub-rectangle for each term in the expansion of \((1 + 0.05) \times (1 + 0.03)\). Their combined area of roughly \(1 + 0.05 + 0.03\) represents an 8% fractional increase over the big part. The big part is 21, and 8% of it is 1.68, so \(3.15 \times 7.21 = 22.68\), which is within 0.14% of the exact product.

Question

Can you find a picture for the correction factor?

Multiple problems

Problem 5.6 Picture for the fractional error

What is the pictorial explanation for the fractional error of roughly 0.15%?

Problem 5.7 Try it yourself

Estimate 245×42 by rounding each factor to a nearby multiple of 10, and compare this big part with the exact product. Then draw a rectangle for the correction factor, estimate its area, and correct the big part.

Low-Entropy Expressions

The correction to \(3.15 \times 7.21\) was complicated as an absolute or additive change but simple as a fractional change. This contrast is general. Using the additive correction, a two-factor product becomes

\[(x + Δx)(y + Δy) = xy + \underbrace{xΔy + yΔx + ΔxΔy.}_{additive correction} \label{5.5} \]

Problem 5.8 Rectangle picture

Draw a rectangle representing the expansion

\[(x + Δx)(y + Δy) = xy + xΔy + yΔx + ΔxΔy.\label{5.6} \]

When the absolute changes \(Δx\) and \(Δy\) are small (\(x ≪ Δx\) and \(y ≪ Δy\)), the correction simplifies to \(xΔy + yΔx\), but even so it is hard to remember because it has many plausible but incorrect alternatives. For example, it could plausibly contain terms such as \(ΔxΔy, xΔx,\) or \(yΔy\). The extent of the plausible alternatives measures the gap between our intuition and reality; the larger the gap, the harder the correct result must work to fill it, and the harder we must work to remember the correct result.

Such gaps are the subject of statistical mechanics and information theory [20, 21], which define the gap as the logarithm of the number of plausible alternatives and call the logarithmic quantity the entropy. The logarithm does not alter the essential point that expressions differ in the number of plausible alternatives and that high-entropy expressions [28]—ones with many plausible alternatives—are hard to remember and understand.

In contrast, a low-entropy expression allows few plausible alternatives, and elicits, “Yes! How could it be otherwise?!” Much mathematical and scientific progress consists of finding ways of thinking that turn high-entropy expressions into easy-to-understand, low-entropy expressions.

Question

What is a low-entropy expression for the correction to the product \(xy\)?

A multiplicative correction, being dimensionless, automatically has lower entropy than the additive correction: The set of plausible dimensionless expressions is much smaller than the full set of plausible expressions.

The multiplicative correction is \((x + Δx)(y + Δy)/xy\). As written, this ratio contains gratuitous entropy. It constructs two dimensioned sums \(x+Δx\) and \(y+Δy\), multiplies them, and finally divides the product by \(xy\). Although the result is dimensionless, it becomes so only in the last step. A cleaner method is to group related factors by making dimensionless quantities right away:

\[\begin{align}  \frac{(x + Δx)(y + Δy)}{xy} &= \dfrac{x + Δx)}{x} \dfrac{(y+Δy)}{y} \\[4pt] &= (1 + \dfrac{Δx}{x}) (1 + \dfrac{Δy}{y}). \label{5.7} \end{align} \]

The right side is built only from the fundamental dimensionless quantity 1 and from meaningful dimensionless ratios: \((Δx)/x\) is the fractional change in x, and \((Δy)/y\) is the fractional change in \(y\).

The gratuitous entropy came from mixing \(x + Δx, y + Δy, x,\) and \(y\) willy nilly, and it was removed by regrouping or unmixing. Unmixing is difficult with physical systems. Try, for example, to remove a drop of food coloring mixed into a glass of water. The problem is that a glass of water contains roughly \(10^{25}\) molecules. Fortunately, most mathematical expressions have fewer constituents. We can often regroup and unmix the mingled pieces and thereby reduce the entropy of the expression.

Problem 5.9 Rectangle for the correction factor

Draw a rectangle representing the low-entropy correction factor

\[\left(1 + \frac{Δx}{x}\right) \left(1 + \frac{Δy}{y}\right).\label{5.8} \]

A low-entropy correction factor produces a low-entropy fractional change:

\[\begin{align}  \frac{Δ(xy)}{xy} &= \left(1 + \dfrac{Δx}{x}\right) \left(1 + \dfrac{Δy}{y}\right) - 1 \\[4pt] &= \dfrac{Δx}{x}+ \dfrac{Δy}{y} + \dfrac{Δx}{x}\dfrac{Δy}{y}, \label{5.9} \end{align} \]

where \(Δ(xy)/xy\) is the fractional change from \(xy\) to \((x + Δx)(y + Δy)\). The rightmost term is the product of two small fractions, so it is small compared to the preceding two terms. Without this small, quadratic term,

\[\frac{Δ(xy)}{xy} ≈ \frac{Δx}{x} + \frac{Δy}{y}. \label{5.10} \]

Small fractional changes simply add!

This fractional-change rule is far simpler than the corresponding approximate rule that the absolute change is \(xΔy + yΔx\). Simplicity indicates low entropy; indeed, the only plausible alternative to the proposed rule is the possibility that fractional changes multiply. And this conjecture is not likely: When \(Δy = 0\), it predicts that \(Δ(xy) = 0\) no matter the value of \(Δx\) (this prediction is explored also in Problem 5.12).

Multiple problems

Problem 5.10 Thermal expansion

If, due to thermal expansion, a metal sheet expands in each dimension by 4%, what happens to its area?

Problem 5.11 Price rise with a discount

Imagine that inflation, or copyright law, increases the price of a book by 10% compared to last year. Fortunately, as a frequent book buyer, you start getting a store discount of 15%. What is the net price change that you see?

Squaring

In analyzing the engineered and natural worlds, a common operation is squaring a special case of multiplication. Squared lengths are areas, and squared speeds are proportional to the drag on most objects (Section 2.4):

\[F_{d} ∼ ρv^{2}A, \label{5.11} \]

where \(v\) is the speed of the object, A is its cross-sectional area, and ρ is the density of the fluid. As a consequence, driving at highway speeds for a distance \(d\) consumes an energy

\[E = F_{d}d ∼ ρAv^{2}d. \nonumber \]

Energy consumption can therefore be reduced by driving more slowly. This possibility became important to Western countries in the 1970s when oil prices rose rapidly (see [7] for an analysis). As a result, the United States instituted a highway speed limit of 55 mph (90 kph).

Question

By what fraction does gasoline consumption fall due to driving 55 mph instead of 65 mph?

Solution

A lower speed limit reduces gasoline consumption by reducing the drag force \(ρAv^{2}\) and by reducing the driving distance \(d\): People measure and regulate their commuting more by time than by distance. But finding a new home or job is a slow process. Therefore, analyze first things first assume for this initial analysis that the driving distance d stays fixed (then try Problem 5.14).

With that assumption, E is proportional to \(v^{2}\), and

\[\frac{ΔE}{E} = 2 x \frac{Δv}{v}. \label{5.12} \]

Going from 65 mph to 55 mph is roughly a 15% drop in \(v\), so the energy consumption drops by roughly 30%. Highway driving uses a significant fraction of the oil consumed by motor vehicles, which in the United States consume a significant fraction of all oil consumed. Thus the 30% drop substantially reduced total US oil consumption.

Multiple problems

Problem 5.12 A tempting error

If A and \(x\) are related by \(A = x^{2}\), a tempting conjecture is that

\[\frac{ΔA}{A} ≈ (\frac{Δx}{x})^{2}. \label{5.13} \]

Disprove this conjecture using easy cases (Chapter 2).

Problem 5.13 Numerical estimates

Use fractional changes to estimate \(6.3^{3}\). How accurate is the estimate?

Problem 5.14 Time limit on commuting

Assume that driving time, rather than distance, stays fixed as highway driving speeds fall by 15%. What is the resulting fractional change in the gasoline consumed by highway driving?

Problem 5.15 Wind power

The power generated by an ideal wind turbine is proportional to \(v^{3}\) (why?). If wind speeds increase by a mere 10%, what is the effect on the generated power? The quest for fast winds is one reason that wind turbines are placed on cliffs or hilltops or at sea.