Appendix Appendix A

Notation, Brick by Brick

Every symbol in this book, in the order it is built, with the place it is built. Nothing here is assumed; everything is constructed on the page listed.

• Multiplication (\(3 \times 4\); this book always writes \(\times\)): the ordinary product. Choices multiply (Chapter 1, Definition 1.1).
• Powers (\(10^4\), \(2^n\)): the little number counts how many copies are multiplied. \(10^4 = 10 \times 10 \times 10 \times 10\); \(2^n\) is \(n\) twos. Built in §1.2 and §1.3.
• The doubling ladder: the table \(2^1 = 2\) through \(2^{20} = 1{,}048{,}576\) (§1.3); extended landmarks in Appendix B.
• Factorial (\(n!\)): the product of all whole numbers from \(n\) down to 1; counts arrangements of \(n\) distinct things. Built in Definition 1.2.
• Falling product (\(n \times (n-1) \times \cdots \times (n-k+1) = \frac{n!}{(n-k)!}\)): ordered arrangements of \(k\) from \(n\). Built in §1.4.
• Choose (\(\binom{n}{k}\), "\(n\) choose \(k\)"): ways to pick \(k\) from \(n\) when order doesn't matter; the falling product divided by \(k!\). Built in Definition 1.3.
• The under-brace (\(\underbrace{\cdots}_{\text{label}}\)): a brace under a repeated pattern, the label saying how many — "twelve copies" in §1.6.
• Probability as a fraction (\(1/2\), \(1/N\)): favorable over possible when outcomes are equally likely; said aloud as "1 in 2," "1 in \(N\)." Built in Definition 2.1.
• Percent and approximately (\(\approx\)): ordinary share-of-a-hundred, and "close enough that the difference doesn't change the point."
• Scientific notation (\(10^{21}\); \(10^{-6}\)): a 1 followed by that many zeros; a minus sign means "one over" — \(10^{-6}\) is \(1/10^{6}\). Glossed where first cosmic (§8.2).
• Flip-count of a number: how many doublings reach it — what power of 2 it is. Named for the record as the base-2 logarithm; computed as math.log2. Built in §4.2.
• Bits of entropy: the flip-count of the space a choice is drawn from (uniform case, Definition 4.1); the expected surprise (general case, Definition 4.2); the flip-count of \(1\) over the likeliest outcome (min-entropy, Definition 5.1).
• The arrow (\(\to\)): "carries to" — \(1/2 \to 1\) bit reads "a one-half chance carries to one bit" (§4.3).
• \(p_{\max}\): the chance of the most likely outcome (Definition 5.1).
• The signature cast (\(d\), \(k\), \(r\), \(s\), \(z\)): private key, nonce, public part of the signature, signature value, message hash — ordinary letters for a toy equation, \(s = (z + r\,d)/k\). Used in Chapter 10 only, with ordinary division standing in for the clock-face kind.
• The double arrow (\(\Longrightarrow\)): "therefore the next line follows" (Chapter 10).