Powers, Roots, and Big Numbers

Take a sheet of paper and fold it in half. You now have 2 layers. Fold it again: each layer doubles, so you have 4. Fold a third time: 8 layers.

Write out what just happened: 2 × 2 × 2 = 8. Step by step: 2 × 2 = 4, then 4 × 2 = 8.

Mathematicians got tired of writing the same number over and over, so they invented a shorthand: a small raised number that counts the copies. On paper it looks like 2 with a tiny 3 floating above it. On a keyboard it is typed 2^3, using the ^ symbol. Both mean exactly one thing: "multiply three copies of 2 together."

The big number is called the base. The small raised number is called the exponent, or the power. So 2^3 = 8 is read "two to the power three equals eight," and it is nothing deeper than three doublings.

Dice work the same way. One die has 6 faces. Roll two dice, and every face of the first can pair with every face of the second: 6 × 6 = 36 combinations, which is 6^2. Roll three dice: 6 × 6 × 6. Work it digit by digit: 6 × 6 = 36, then 36 × 6 is 30 × 6 = 180 plus 6 × 6 = 36, and 180 + 36 = 216. So 6^3 = 216. Count the copies, and you have the exponent.

There is an old story about a king who agreed to a reward that sounded tiny: 1 grain of wheat on the first square of a chessboard, 2 on the second, 4 on the third — doubling on each of the 64 squares.

Follow the doublings yourself: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1,024. That last number is 2^10. Ten squares in, the king already owes more than a thousand grains.

Now push further. 2^20 is 2^10 × 2^10, which is 1,024 × 1,024. Work it in two pieces: 1,024 × 1,000 = 1,024,000, and 1,024 × 24 = 24,576. Add them: 1,024,000 + 24,576 = 1,048,576 — past a million after twenty squares. By square 31 the king owes over a billion grains. Square 64 alone holds 2^63 grains: 9,223,372,036,854,775,808 — about nine billion billion, more wheat than the world has ever grown. The king lost his kingdom to a doubling.

Another version of the same surprise: a sheet of paper is about 0.1 millimeters thick. If you could fold it in half 42 times, its thickness would be 0.1 mm × 2^42 — roughly 440,000 kilometers, which is past the Moon (about 384,000 km away). To be honest: nobody can physically fold paper more than about seven times. The math outruns the paper almost immediately. That is the point.

Numbers this size are clumsy to write, so there is a shorthand for them too: scientific notation. 10^3 means 10 × 10 × 10 = 1,000 — a 1 followed by 3 zeros. So 9,223,372,036,854,775,808 is written 9.2 × 10^18: "take 9.2 and slide the decimal point 18 places to the right." When you see 10^18, read it as "an 18-zero number" and move on. The exponent tells you the size at a glance.

Squaring a number means multiplying it by itself once: 5 squared is 5 × 5 = 25, written 5^2. The name is literal — a square floor that is 5 tiles by 5 tiles contains 25 tiles.

A square root undoes that. It asks: which number, times itself, gives 25? The answer is 5. The symbol for this question is √, called the radical sign: √25 = 5. A few more, worth checking on a calculator: √9 = 3 because 3 × 3 = 9. √16 = 4 because 4 × 4 = 16. √100 = 10.

Most square roots are not tidy. √2 = 1.41421356... — the digits go on forever, and a calculator simply gives you as many as you need.

Here is the one that matters most in this curriculum: which number, times itself, gives 0.5 (one half)? The answer is about 0.7071. Check it by hand. Ignore the decimal points and multiply 7,071 × 7,071 in three pieces: 7,071 × 7,000 = 49,497,000. Then 7,071 × 70 = 494,970. Then 7,071 × 1 = 7,071. Add them: 49,497,000 + 494,970 = 49,991,970, and 49,991,970 + 7,071 = 49,999,041. Now restore the decimal points — four decimal digits in each factor means eight in the answer: 0.7071 × 0.7071 = 0.49999041. A hair under 0.5. Keep more digits (0.70710678...) and the square lands as close to 0.5 as you like. You will meet 0.7071 in nearly every quantum circuit you ever run.

One more thing worth staring at: squaring a number smaller than 1 makes it smaller. 0.5 × 0.5 = 0.25 — half of a half is a quarter. Keep that in your pocket. Quantum probabilities come from squaring numbers like 0.7071, and this shrinking is why the arithmetic works out.

A bit is the smallest piece of information there is: a single symbol that is either 0 or 1. A light switch, up or down.

Put two bits side by side and list every string you can write: 00, 01, 10, 11. Four strings — which is 2^2, because the first bit can be written 2 ways and, for each of those, the second bit can be written 2 ways: 2 × 2 = 4.

Three bits? Take each of the four strings above and append a 0 or a 1 — every string splits into two. The full list: 000, 001, 010, 011, 100, 101, 110, 111. Eight strings, and 8 = 2^3. Every extra bit doubles the list. This is the chessboard again.

A quantum computer with n qubits prints one string of n bits every time you measure it. So n qubits have 2^n possible readouts: 3 qubits have 8, 10 qubits have 1,024, and 50 qubits have 2^50 — about 1.1 × 10^15, a million billion. This doubling is why a machine you could hold in a diagram describes a list of possibilities no notebook could hold.

One reading convention before you run anything. Qubits are numbered 0, 1, 2, ... and in every readout string here, the rightmost character is qubit 0 — just like ordinary numbers keep the ones place on the right. So the string 110 means: qubit 2 reads 1, qubit 1 reads 1, qubit 0 reads 0.

Everything in this chapter gets used, soon and often:

• 2^n readouts is the first fact of quantum computing. You will see it again the moment you meet real qubits in Bits vs qubits and when machines grow in Multiple qubits. For a two-minute refresher any time, there is qubit vs bit.
• 0.7071 and squaring return in the very next chapter, Probability without formulas, and then everywhere in Probability amplitudes.
• Lists with 2^n entries are the subject of Lists, vectors, and normalization.
• Exponents running downhill (0.99 multiplied by itself 100 times) decide whether a real quantum computer works at all — that story is Exponentials, logs, and decay.

And the circuit above is yours to keep: open it in the Lab, add a third X, and check that you can predict the string before every run.