Place Value & Number Systems

Grades 2–5 Place value Bases

We only have ten digits (0–9) but infinitely many numbers. The trick that makes it work: a digit's value depends on where it sits. The 4 in 467 isn't "four" — it's four hundreds. Position is meaning.

4 6 7 ↑ ↑ ↑ 4×100 6×10 7×1 400 + 60 + 7 = 467

Everything in arithmetic rides on this. Carrying is just "ten of this place become one of the next place." Borrowing is the reverse. Both directions of our addition and subtraction methods are place value in motion.

Base 10 isn't the only way

Ten is a habit, not a law — we probably use it because of fingers. Any number can be the base. Base 2 (binary) needs only two digits, 0 and 1, and each place is worth ×2 instead of ×10:

binary: 1 0 1 1 ↑ ↑ ↑ ↑ 8s 4s 2s 1s 1×8 + 0×4 + 1×2 + 1×1 = 11 (that's eleven in decimal!)

Base 10: digits 0–9, places worth 1, 10, 100, 1000…
Base 2 (binary): digits 0–1, places worth 1, 2, 4, 8, 16… — the native language of every computer.
Base 16 (hexadecimal): digits 0–9 then A–F — how programmers write colors and memory addresses.
Base 60: ancient Babylon — the reason hours split into 60 minutes.

The idea that changed computing Claude Shannon's master's thesis showed that binary arithmetic and yes/no logic are the same thing — which is why all computers run on 1s and 0s. It is routinely called the most important master's thesis ever written:

Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27(3–4), 379–423 & 623–656. doi:10.1002/j.1538-7305.1948.tb01338.x — the paper (on IEEE Xplore) that founded information theory.

See it in action on the math inside computers page.

Try it yourself:

What is 1001 in binary worth in decimal? (8 + 1 = 9)
Write 6 in binary (110)
In 5,832 what is the 8 worth? (800)

Next: scientific notation is place value stretched to the biggest and smallest numbers in the universe.

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