Math Toolkit

Pythagoras

In a right triangle, the squares on the two short sides (the legs) add up to the square on the long side (the hypotenuse). Legs 3 and 4: \(9 + 16 = 25\), so the long side is \(\sqrt{25} = 5\).

\[ \begin{aligned} a^2 + b^2 &= c^2, \\[4pt] c &= \sqrt{a^2 + b^2}, \\[4pt] b &= \sqrt{c^2 - a^2}. \end{aligned} \]

The same rule measures the straight-line distance between two points on a map: the east offset and the north offset are the legs.

Practice

Sine, cosine and angles

In a right triangle, pick one of the two sharp angles and call it \(\theta\). The side that touches it (other than the hypotenuse) is the adjacent side; the side across from it is the opposite side. Their ratios to the hypotenuse depend only on the angle, not on how big the triangle is:

\[ \begin{aligned} \cos\theta &= \frac{\text{adjacent}}{\text{hypotenuse}}, \\[6pt] \sin\theta &= \frac{\text{opposite}}{\text{hypotenuse}}, \\[6pt] \tan\theta &= \frac{\text{opposite}}{\text{adjacent}}. \end{aligned} \]

Some people remember this as SOH-CAH-TOA. In the 3-4-5 triangle, the angle next to the side of length 4 has \(\cos\theta = 4/5 = 0.8\) and \(\sin\theta = 3/5 = 0.6\). That angle is about 36.87°. Because the two legs are \(\cos\theta\) and \(\sin\theta\) times the hypotenuse, Pythagoras gives \(\cos^2\theta + \sin^2\theta = 1\) for every angle.

Degrees, radians and your calculator

Angles come in two units. In degrees a full turn is 360°; in radians it is \(2\pi \approx 6.283\). This course uses degrees. If your calculator says \(\cos 60 = -0.952\) instead of 0.5, it is set to radians: switch it to degrees (look for a DRG or MODE key). Code uses radians too: see Code basics.

Going backwards: inverse cosine

If you know the ratio and want the angle, use the inverse cosine, \(\arccos\). Calculators label it \(\cos^{-1}\); it means "the angle whose cosine is", not \(1/\cos\). So \(\arccos 0.8 = 36.87°\) and \(\arccos 0.6 = 53.13°\). Inverse sine, \(\arcsin\), works the same way.

Practice

Squares and square roots

Squaring a number multiplies it by itself: \(7^2 = 49\). The square root undoes that: \(\sqrt{49} = 7\). Picture a square made of unit tiles. Its area is the square; its side is the square root.

Two rules carry most of the course. Roots of products split: \(\sqrt{4 \times 9} = \sqrt{4}\,\sqrt{9} = 6\). Roots of sums don’t: \(\sqrt{9 + 16} = 5\), not \(3 + 4 = 7\). And a negative number has no square root among ordinary numbers, which is why \(\sqrt{1 - v^2}\) needs \(v\) between −1 and 1 (that is, \(v^2\) below 1).

Practice

Slopes and graphs

The slope of a straight line is how much it rises for each step it runs: rise ÷ run. A line through (0, 0) and (4, 2) rises 2 while it runs 4, so its slope is \(2/4 = 0.5\). A line written \(y = m x + b\) has slope \(m\) and crosses the vertical axis at \(b\).

In a physics class, distance–time graphs put time along the bottom. A spacetime diagram puts time \(t\) pointing up, and distance \(x\) pointing right. A point on the diagram is an event, a time and a place, and the course writes it with time first: \((t, x)\). The path an object traces on the diagram is its worldline.

Because time points up, a moving object’s speed is how far it goes sideways for each step up: \(v = \Delta x / \Delta t\). That is run ÷ rise, the flip of the usual slope. So a steep line is slow, a vertical line is standing still, and a 45° line is light, which goes one step sideways for each step up.

Practice

Units as ratios

“60 minutes per hour” is a fraction equal to 1: \(\frac{60\ \text{min}}{1\ \text{h}} = 1\). Multiplying by 1 never changes a quantity, only its units. So to convert, multiply by fractions like that, arranged so the unit you don’t want cancels top against bottom. The top row of Toolkit 5 turns 3 hours into seconds this way.

Light’s speed is a conversion factor

The course’s most important conversion factor is light’s speed. Light covers exactly 299 792 458 meters every second. That lets the course measure time in meters. One meter of time is the time light takes to go one meter: 1 ÷ 299 792 458 of a second, about 3.3 nanoseconds (billionths of a second).

So 1 second and 299 792 458 meters of time are two names for the same amount of time, like 1 hour and 60 minutes. So this fraction equals 1, the same way \(\frac{60\ \text{min}}{1\ \text{h}}\) does:

\[ \frac{299\,792\,458\ \text{m of time}}{1\ \text{s}} = 1. \]

Multiplying by it turns seconds into meters of time without changing how much time you have.

In 1 meter of time, light goes 1 meter. So light’s speed is 1 m (of distance) ÷ 1 m (of time) = 1, with no units left over. Slower things get smaller numbers: a car covers only a tiny part of a meter in each meter of time.

Measure time in meters and every speed becomes a plain number, the fraction of light’s speed. The course calls it \(v\). Here is the move in both directions:

• \(v = 0.6\) means 0.6 × 299 792 458, about 180 000 000 meters per second.
• A car at 30 m/s has \(v\) = 30 ÷ 299 792 458, about 0.000 000 1. The bottom row of Toolkit 5 shows every unit cancelling.

Practice

Rearranging equations

An equation is a balance. Do the same thing to both sides and it stays balanced. To get a letter alone, undo whatever was done to it, in reverse order: if it was multiplied by 2 and then had 1 added, subtract 1 first, then divide by 2.

The course’s favorite rearrangement involves \(\gamma\) (gamma), the stretch factor: the number that says how big relativity’s effects are at a speed \(v\) (\(\gamma = 1\) means no effect at all). Its formula, \(\gamma = 1/\sqrt{1 - v^2}\), run backward gives the speed, \(v = \sqrt{1 - 1/\gamma^2}\). Each step is something you already know: flip both sides, square both sides, move a term across, take a root.

Practice

Fractions

The course’s friendliest speeds are fractions: 3/5, 4/5, 5/13, 12/13. Working with them exactly keeps the answers clean. Three moves cover almost everything. To add, use a common bottom: \(\tfrac{3}{5} + \tfrac{4}{5} = \tfrac{7}{5}\). To divide by a fraction, multiply by its flip: \(1 \div \tfrac{4}{5} = \tfrac{5}{4}\). To square, square the top and the bottom: \(\left(\tfrac{3}{5}\right)^2 = \tfrac{9}{25}\).

Practice

Exponentials and logarithms

The number \(e \approx 2.718\) is the natural base for growth and decay. \(e^x\) means \(e\) multiplied by itself \(x\) times (and it makes sense for any \(x\), not only whole numbers). The natural logarithm \(\ln\) undoes it: \(\ln(e^x) = x\), and \(e^{\ln y} = y\).

The rule that makes logarithms useful: they turn multiplying into adding, \(\ln(a\,b) = \ln a + \ln b\). Decay uses \(e\) too: for muons sitting still in a lab, the fraction still alive after a time \(t\) is \(e^{-t/\tau}\), where \(\tau\) is the average lifetime, 2.2 μs. For fast muons, the key question is: whose clock measures \(t\)? A later chapter answers it.

Practice

Approximations for small numbers

When a number is tiny, some awkward formulas shrink to arithmetic you can do in your head. Take \(x\) = 0.01. Then 1 − \(x\) = 0.99, and 1 ÷ 0.99 = 1.0101…, very nearly 1 + \(x\) = 1.01. And 1 ÷ √0.99 = 1.00504…, very nearly 1 + \(x\)/2 = 1.005. The pattern holds for any tiny number \(x\):

\[ \begin{aligned} \frac{1}{1 - x} &\approx 1 + x, \\[8pt] \frac{1}{\sqrt{1 - x}} &\approx 1 + \frac{x}{2}. \end{aligned} \]

Two cousins work the same way. Flip the sign: \(1/(1 + x) \approx 1 - x\) (1 ÷ 1.01 = 0.990…). Square: \((1 + x)^2 \approx 1 + 2x\) (1.01² = 1.0201).

What it means for speeds

The stretch factor \(\gamma\) (the Greek letter gamma) is the number that says how big relativity’s effects are at a speed \(v\) (\(\gamma = 1\) means no effect at all). Its formula is \(\gamma = 1/\sqrt{1 - v^2}\), and Chapter 5 builds it from a clock made of light. Put \(x = v^2\) into the second shortcut and you get the course’s most useful one:

\[ \gamma \approx 1 + \tfrac12 v^2. \]

So the car’s \(\gamma\) is 1.000 000 000 000 005, and a 10-digit calculator shows 1. A jet airliner at 250 m/s has \(v\) of about 0.000 000 83, and half its square is about 0.000 000 000 000 35. That is why relativity hides at the speed of a car or a plane: it is there, 13 or more places after the decimal point. The shortcut works only for small \(v\): at \(v = 0.995\) it gives 1.5, but the true \(\gamma\) is 10.

Practice

cosh, sinh and tanh

Chapter 13 builds these functions and uses them for speeds. If no chapter has sent you here yet, it is fine to skip this section for now.

These three functions sound exotic, but they are built from \(e^x\) with plain arithmetic:

\[ \begin{aligned} \cosh\theta &= \frac{e^{\theta} + e^{-\theta}}{2}, \\[6pt] \sinh\theta &= \frac{e^{\theta} - e^{-\theta}}{2}, \\[6pt] \tanh\theta &= \frac{\sinh\theta}{\cosh\theta}. \end{aligned} \]

Try \(\theta = \ln 2\), so \(e^{\theta} = 2\) and \(e^{-\theta} = \tfrac12\). Then \(\cosh\theta = (2 + 0.5)/2 = 1.25\), \(\sinh\theta = (2 - 0.5)/2 = 0.75\), and \(\tanh\theta = 0.75/1.25 = 0.6\). All three belong to the speed \(v = 0.6\) (3/5): 0.6 is \(v\) itself, 1.25 is its stretch factor \(\gamma\), and 0.75 is \(\gamma v\), a number a later chapter puts to work.

Now check: \(1.25^2 - 0.75^2 = 1.5625 - 0.5625 = 1\). That is no coincidence: \(\cosh^2\theta - \sinh^2\theta = 1\) for every \(\theta\). So the point \((\cosh\theta, \sinh\theta)\) always lies on the hyperbola \(x^2 - y^2 = 1\), the same way \((\cos\phi, \sin\phi)\) always lies on the circle \(x^2 + y^2 = 1\).

Practice

Code basics

Every “Build it” section asks for one small JavaScript function. Here is everything the first one needs, for reference. Later chapters add a little more, and each addition is explained right where it’s first used.

• A function has a name, inputs in brackets (parameters), and a body in braces. return sends back the answer.
• Arithmetic: + - * /. Powers: 3 ** 2 is 9. Brackets work as in algebra: (1 + 2) * 3 is 9.
• Square roots: Math.sqrt(49) is 7. Pythagoras in one call: Math.hypot(3, 4) is 5.
• Scientific notation: 3e8 is \(3 \times 10^8\) and 2.2e-6 is \(2.2 \times 10^{-6}\).
• Angles: Math.cos and Math.sin use radians, not degrees, so convert first: Math.cos(60 * Math.PI / 180) gives 0.5 (to 15 decimal places). Going backwards, Math.acos(0.8) gives radians (0.6435); multiply by 180 / Math.PI to get degrees (36.87).
• The test table lists each test’s name (what it checks), the Input it tried, the Expected answer, what your function returned (Got) and the Result. Fix the first failing row first. For that row, compare Got with Expected for the same Input.

Here is a made-up table for double(x), written by mistake as return x + 2;.

The first row passes by luck: 2 + 2 and 2 × 2 are both 4. The second row fails. For the input 21, Expected says 42 but Got says 23, which is 21 + 2. So the function adds 2 where it should multiply by 2.

Practice

Ready to use it? The prologue’s Build it section has a one-line warm-up and your first function.