Bases & Adics

Chapter \(1_{\,2}\): what base 10 really is (and why \(2+2\) can look like \(10\))

We write numbers using places. In base \(10\) there are ten symbols: \(0,1,2,3,4,5,6,7,8,9\). Each place is a power of \(10\).

• The rightmost place means “ones.”
• Next left means “tens.”
• Next left means “hundreds,” and so on.

So \(745\) means 7 hundreds, 4 tens, 5 ones: \(7\cdot 100 + 4\cdot 10 + 5\).

The idea of a base is to pick a bundle size. In base \(10\), ten ones bundle into a ten, ten tens into a hundred. If you change the bundle size to \(b\), the allowed digits are \(0\) up to \(b-1\). Each place is a power of \(b\). That’s the only change.

• Base \(4\) uses digits \(0,1,2,3\) and places \(1,4,16,64,\dots\).
• Base \(2\) uses digits \(0,1\) and places \(1,2,4,8,\dots\).

When someone says “\(2+2 = 10\),” they’re talking about notation, not value. The value four is written “\(10\)” in base \(4\) (one group of four and zero ones). In base \(2\), four is “\(100\)”.

Rule: inside one line, stick to one base. In base \(2\) you can’t even write the symbol “2,” so “\(2+2\)” isn’t legal there. You’d write \(10+10=100\) to mean “two plus two equals four” in base 2.

How carrying and borrowing work in any base

Carrying means you hit the bundle size and trade. In base \(b\), when a column sums to \(\ge b\), write the remainder and carry \(1\) to the next column.

Multiplication and division follow the same school algorithms; the carry/borrow threshold is \(b\) instead of \(10\).

Chapter \(10_{\,2}\): reading & writing across bases

Read a base-\(b\) string

Expand by powers of \(b\).

Write a base-\(b\) string for a whole number \(N\)

Keep dividing by \(b\) and record the remainders; read them bottom to top.

Fractions

Multiply the fractional part by \(b\) repeatedly; each whole part is the next digit after the point.

Rationals terminate or repeat in any base. If the denominator’s primes are all in the base, you get a terminating expansion (e.g. base \(10\): only \(2\)s and \(5\)s terminate); otherwise it repeats.

Quick conversions when bases are powers

• Binary → octal: group bits in 3s from the point.
• Binary → hex: group bits in 4s.

Sanity checks

• Value check: convert small samples to base \(10\).
• Digit range: digits must be \(0,\dots,b-1\).
• Length check: huge values shouldn’t have suspiciously few digits.

All of this is still regular place value. Now we flip the viewpoint.

Chapter \(11_{\,2}\): a new sense of “close,” and 10-adic intuition

In adic land, “small” means “divisible by a big power of the base.” Two integers are close if they agree in many trailing digits.

• Match last digit ⇔ difference is multiple of \(10\).
• Match last two digits ⇔ multiple of \(100\).
• Match last \(k\) digits ⇔ multiple of \(10^k\).

Define distance so more shared trailing digits means closer. Powers of \(10\) become tiny steps. A 10-adic integer pins down every finite tail of decimal digits—picture a left-infinite \(\cdots d_3 d_2 d_1 d_0\). School arithmetic works; carries run left, and that’s fine because each finite tail stabilizes.

Catch: \(10\) isn’t prime, so 10-adics split into \(2\)-adics × \(5\)-adics (zero divisors). For clean algebra, use a prime \(p\): the \(p\)-adics.

Chapter \(100_{\,2}\): \(p\)-adics from the ground up

Divisibility count & size

Let \(v_p(n)\) be the exponent of \(p\) dividing nonzero \(n\). Bigger \(v_p\) ⇒ “smaller” in \(p\)-adic size. Completing \(\mathbb Q\) under this size gives \(\mathbb Q_p\); its “integers” are \(\mathbb Z_p\).

Three equivalent views of \(\mathbb Z_p\)

• Limits of integers agreeing on longer base-\(p\) tails.
• Inverse limit of \(\mathbb Z/p^k\mathbb Z\).
• Power series \(a_0+a_1p+a_2p^2+\cdots\) with \(a_i\in\{0,\dots,p-1\}\).

Arithmetic is school arithmetic on digits, carrying left.

Units & why prime matters

In \(\mathbb Z_p\), numbers not divisible by \(p\) are invertible (first digit \(a_0\neq 0\)). Prime \(p\) ⇒ no zero divisors; the fraction field \(\mathbb Q_p\) is a field.

Compact notation & ultrametric

\(|x|_p = p^{-v_p(x)}\) (and \(|0|_p=0\)), distance \(d_p(x,y)=|x-y|_p\). Triangle rule: \(|x+y|_p \le \max(|x|_p,|y|_p)\).

By hand

• Negatives: \(-1=\cdots (p-1)\).
• Geometric: \(1+p+p^2+\cdots=\frac{1}{1-p}\).
• Inverses by lifting (when \(p\nmid m\)): solve one digit at a time via modular inverses.

Chapter \(101_{\,2}\): connecting pictures & quick practice

Three views, one object

• Digits (great for hand arithmetic)
• Residues mod \(p^k\) (great for lifting)
• Metric/limits (great for convergence/analysis)

Chinese remainder: composite bases (e.g. \(10=2\cdot 5\)) split into prime pieces, introducing zero divisors—hence the focus on \(p\) prime.

Units, valuation, and size

• \(v_p(x+y)\ge \min(v_p(x),v_p(y))\) (strict if leading digits cancel).
• \(|p|_p=1/p\): powers of \(p\) shrink.
• \(a/b\in\mathbb Z_p \iff v_p(a)\ge v_p(b)\).

Negatives & repeating tails

• In base \(p\), \(\cdots (p-1) = -1\).
• More generally, left repeats encode negatives plus a finite correction.

When Hensel can fail

If \(f'(x_0)\equiv 0\pmod p\), the simple version doesn’t apply (there are refined forms).

Worked examples you can trace

Practice (tap to reveal answers)