MISSION · Number sense
Comparing and Ordering Numbers
The first digit that differs settles it
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LEVEL 2 · LEARN
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The idea in one sentence
To compare two numbers, line them up by place value and read from the left — the first column where they differ decides everything.
Why left to right
Compare 472 and 439. Start at the hundreds: both have 4, so no decision yet. Move to the tens: 7 beats 3. Stop. 472 is bigger, and nothing in the ones column can change that.
That’s because a single ten outweighs any number of ones you could have, and a single hundred outweighs any number of tens. Each place is worth more than everything to its right put together, so the leftmost difference wins.
It also means a longer number is bigger, as long as there are no leading zeros. 1,000 beats 999 without comparing any digits at all, because it has a thousands column and the other doesn’t.
The symbols
\(>\) means greater than, \(<\) means less than, and \(=\) means equal to.
\[472 > 439 \qquad 8 < 15 \qquad 20 = 20\]The open end always faces the bigger number and the point faces the smaller one. Some people picture a crocodile eating the larger meal; others notice that \(<\) looks a little like the letter L for less. Either works — what matters is reading the whole statement in order, left to right, so \(8 < 15\) is “eight is less than fifteen.”
The number line settles arguments
Put the numbers on a line and comparison becomes position: whatever sits further right is bigger. That picture keeps working when the numbers get harder.
It’s how you see that \(-5\) is less than \(-2\), which surprises people who compare the digits and think 5 beats 2. On the line, \(-5\) is further left, so it’s smaller. It’s also how \(0.7 > 0.65\) becomes obvious, even though 65 looks like the bigger number.
Whenever a comparison feels slippery, drawing the line resolves it.
Ordering a list
To sort several numbers, compare them in pairs. Find the smallest, set it aside, then find the smallest of what’s left.
Ascending means smallest first, climbing. Descending means largest first. The words turn up in every set of instructions you’ll ever be given, and mixing them up gets a correct answer marked wrong.
Common mix-ups
- Comparing from the right. The ones column never decides anything unless everything to its left already matched.
- Judging by how many digits look big. 91 is smaller than 100, whatever the 9 suggests.
- Pointing the symbol the wrong way. The open side faces the larger number.
- Treating negatives like positives. \(-9\) is less than \(-1\).
Check yourself
Put these in order, smallest first: 307, 37, 370, 73. Then fill in the symbol: \(-4 \; \square \; -6\).
37, 73, 307, 370. And \(-4 > -6\), because \(-4\) sits further right on the number line.
LEVEL 3 · REFLECT
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