A digit’s value is the digit multiplied by its column. In 3,456 the 4 sits in the hundreds, so it is worth 400 — not 4. Written out: 3,000 + 400 + 50 + 6. Each column is worth ten times the one to its right, and that is the whole system.
Ten times the one to its right
Every column is worth ten times its right-hand neighbour. That one rule generates the entire notation, in both directions, forever.
It is why the same ten symbols can write any number at all. Roman numerals need a new symbol as the numbers grow — I, V, X, L, C, D, M — and run out. Place value never does; it just adds a column.
Ten is not special mathematically, only anatomically. Base two uses the same rule with two symbols, base sixteen with sixteen, and computers run on the first of those for reasons that have nothing to do with fingers.
Three expanded forms
“Write it in expanded form” means three different things depending on which year you are in, so all three are on the page.
As a sum: 3,000 + 400 + 50 + 6. The earliest version, and the one that shows the place values directly.
As products: (3 × 1,000) + (4 × 100) + (5 × 10) + (6 × 1). This one separates the digit from its column, which is the idea being taught.
With powers of ten: (3 × 10³) + (4 × 10²) + (5 × 10¹) + (6 × 10⁰). The version that generalises — it is polynomial notation, and a number is a polynomial in ten.
That last observation is not decoration. It is exactly why long multiplication and polynomial multiplication look the same on the page.
What a zero is for
A zero digit is not a placeholder in the dismissive sense. It is doing the single most important job in the notation.
It holds a column open. Without it, 105 and 15 would be written the same way, and the whole system would be ambiguous.
This is why zero arrived so late historically. Systems that wrote numbers without it — and most did — had to leave a gap and hope, or use a separate symbol per magnitude. The decimal place-value system that reached Europe through al-Khwārizmī’s work carried zero with it, and that is what made it work.
The table above names every zero digit’s column, and the expanded sum leaves it out. Both are correct: the column exists and contributes nothing.
The decimal side
The same rule continues past the point: each place is a tenth of the one before it.
The first is tenths, then hundredths, then thousandths. There is no “oneths”, and that is the thing worth stating plainly rather than letting people discover by being wrong.
The reason is that the mirror is around the ones column, not around the decimal point. Tens and tenths sit either side of ones; hundreds and hundredths sit either side of that. The point is a marker for where the ones column ends, not an axis of symmetry.
Which is also why 0.5 and .5 are the same number, and why the leading zero is a readability convention rather than a mathematical requirement.
Where it gets used
Rounding. Rounding to the nearest hundred means looking at the tens digit. Every rounding rule is phrased in place values.
Reading numbers aloud. Grouping in threes exists so that the word form has a rhythm — thousands, millions, billions — and the commas mark where the words change.
Written methods. Column addition, long multiplication and long division all work by keeping place values aligned. Every one of their classic errors is a misaligned column.
Other bases. Binary, octal and hexadecimal are the same idea with a different multiplier, which is why understanding place value once is enough for all of them.
Sources and methodology
Place-value notation and its naming are curriculum standards; these are the references.
Method. The number is decomposed from its digit string rather than by dividing a floating-point value repeatedly by ten, which would introduce errors on the decimal side long before the last place. Word form is built group by group in thousands, so it reads the way a person would say it, and the suite checks it against known values including the teens inside a group — "one thousand thirteen" is the case a naive generator gets wrong. That engine is verified on every change against 140 hand-written assertions, including that the expanded forms sum back to the original number and that a zero digit is named in the table but left out of the expanded sum. The count and the per-case breakdown are published on the formula verification page.
Read the guide
Rounding is a place-value operation more than an arithmetic one — the Rounding Calculator shows which column decides.