The Tens Decide
To compare two-digit numbers, look at the tens first — a whole ten beats any pile of ones — and the signs >, <, and = write down the verdict. · 9 min
Two numbers, one question: which is more, 52 or 47? The 7 in 47 looks big. Do not let it fool you. In a two-digit contest, the tens speak first.
Guess before you learn
Which is more: 52 or 47?
52. If you picked 47 because of that big 7, most people do — the ones are louder, but they are smaller. Five bundles of ten beat four bundles, and seven loose ones can never fill the missing bundle. This folio shows you why.
K–2
3–5
The rule has two steps. First compare the tens; the bigger tens digit wins on the spot. Only when the tens tie do the ones get a vote: 74 and 78 both hold seven tens, so the ones decide, and 78 wins. Why do tens outrank ones? A numeral can carry at most 9 loose ones, and 9 is still less than a single bundle of ten.
Three signs write the verdict. 52 > 47 says '52 is more than 47'. 36 < 63 says '36 is less than 63'. The open side of the sign always faces the bigger number, and = says both sides hold exactly the same.
6–8
This is place-value comparison: start at the highest place and walk right. A lead in a higher place can never be overturned, because the most the lower places can offer — 9 — is less than one step in the higher place — 10. If the tens tie, the ones decide; if every digit ties, the numbers are equal, and = records it.
Read the signs as full sentences. 47 < 52 and 52 > 47 state the same fact from opposite ends. And = is a claim of perfect balance — not a drum-roll before an answer.
9–12
Comparing digit by digit from the left is lexicographic order — dictionary order — and for numerals of equal length it agrees exactly with numeric order, because place values dominate: the largest possible tail is 9, or 99, or 999, always one short of a single step in the next place up. For unequal lengths the dictionary rule alone misfires — it would file 9 after 100 — so compare lengths first, or pad with leading zeros: 009 against 100.
K–2
52 or 47? Count bundles of ten. 52 has five bundles. 47 has four. Five bundles beat four bundles, so 52 is more. The loose ones are too small to change that.
Could the seven loose ones save 47? No. Even nine loose ones are less than one whole bundle. Ones can never catch a missing ten.
Undergrad
Theorem: on fixed-length base-b numerals, lexicographic order coincides with numeric order. Proof: suppose two numerals first differ at a place worth bᵏ. The difference contributed there is at least bᵏ, while all the places below it can differ by at most (b − 1)(bᵏ⁻¹ + … + b + 1) = bᵏ − 1. The higher place wins outright. Unequal lengths reduce to this case by padding with zeros, or by ruling longer-is-larger first — the shortlex order.
The corollary is speed: comparing two n-digit numbers costs at most n digit-comparisons, and usually just one. That efficiency is a dividend of positional notation, not a separate invention.
Postgrad
Canonical base-b numerals under shortlex form an order-isomorphic copy of (ℕ, ≤); pure lex order on arbitrary digit strings fails badly — with 0 < 1 it admits the infinite descent 1 > 01 > 001 > …, so it is not even well-founded. The dominance inequality (b − 1)(bᵏ⁻¹ + … + 1) = bᵏ − 1 < bᵏ is the engine behind radix sort, most-significant-digit key comparison in databases, and every 'compare from the top' algorithm.
The property generalizes: a numeration system admits digitwise comparison precisely when canonical greedy representations are used. Zeckendorf's Fibonacci-weight system compares lexicographically for the same dominance reason. Wherever weights grow fast enough that no tail can overtake a head, 'the tens decide' survives as a theorem.
> and <
The signs that record which number is more. The open side always faces the bigger number; = says both sides match exactly.
There is a picture for this. Give every number a spot: bundles across, loose ones up. Then watch who sits farther right.
Compare 63 and 59 — the steps fade as you master them
63 has 6 tens; 59 has 5 tens
6 tens beat 5 tens — 63 wins
63 > 59
Why is this true?
Why do we bother looking at the ones in 74 and 78?
Because the tens tie at seven each. When the tens say nothing, the ones cast the deciding vote: 78 carries more ones, so 74 < 78.
You now hold the whole contest in two looks: tens first, ones for ties, a sign to write it down. Next folio, counting learns to leap — by twos, fives, and tens.
Practice — new ink and old, interleaved
1.Put these numbers in order from smallest to largest.
- 47
- 39
- 61
- 52
2.Without looking back: what does each digit in 56 count?
The 5 counts bundles of ten — fifty — and the 6 counts loose ones. Fifty-six.
How close were you? Grade yourself honestly — it sets your review date.
3.Without looking back: what do leftovers tell you after pairing, and what does a perfect pairing tell you?
Leftovers sit on the bigger side, so that group has more. A perfect pairing with none left over means the two groups are exactly the same amount.
How close were you? Grade yourself honestly — it sets your review date.
4.Two numbers have the same tens digit. How do you find the bigger one?
When the tens tie, compare the ones: the number with more ones is the bigger one.
How close were you? Grade yourself honestly — it sets your review date.
5.Ten bundles of ten make what number?
6.How many bundles of ten does 83 hold?
7.Pick the true sentence.