# Base three numbers and expanded form

I need help understanding this. Write each of the following base three numerals in expanded notation.

1. $22_3$
2. $212_3$
3. $12110_3$
• What is expanded notation? My guess is something like $123_{10}=1\cdot 10^2+2\cdot 10^1 +3\cdot 10^0$. If so, what is your problem? Mar 12, 2015 at 3:34
• I don't understand what the base three means and expanded notation.
– Alli
Mar 12, 2015 at 3:38
• We don't have a single symbol for every number, so we repeat symbols and use the location of the digit to give us more information. Where in decimal (our usual base 10), the number $234 = 200+30+4=2\cdot 10^2 + 3\cdot 10^1 + 4\cdot 10^0$, the location tells us "how much that digit is worth (in terms of powers of ten)". In a different base, say base $b$, you have $112_b=1\cdot b^2 + 1\cdot b^1 + 2\cdot b^0$ and the location of the digit tells us "how much that digit is worth (in terms of powers of b). So, for base 3, $22_3 = ...$ and $212_3=...$ Mar 12, 2015 at 4:08
• "Expanded notation" is not a term I recognize, so maybe it is defined in your text (if you have one). I took a guess, which I thought was a reasonable one. What do you know about base 3? You should be able to make an analogy from my base 10 answer to base 3(Hint: replace 10s by 3s, but that won't show you understand what base 3 means) Mar 12, 2015 at 4:18

If you are being asked to write numbers in the expanded notation for Base $$3$$ such as $$112201_3$$ (just a random number I made up) then you would need to write it like so (starting from the $$3^5$$th place, the 5th place of the digit)
$$112201_3 = 1.3^5 + 1.3^4 + 2.3^3 + 2.3^2+ 0.3^1 + 1.3^0$$