ATTENTION: To make sure that you can view this blog post properly, make sure that you have the Cambria Math font installed on your computer.
So this is sort of confusing, but it'll make perfect sense after you read this.
So let's first clarify what Base Ten is, it's what we use for counting. If I were to pry off all of the number keys on my keyboard (excluding the ones that are near the Num Lock because those are duplicates of the ones above the letters), then my keyboard would be missing ten keys if you include the zero key.
Basically we're using a number system in which ten is the lowest —amount that's represented by a two digit number (if you don't include anything below one.)— So if I were to keep adding apples to a basket (starting with 0) until I had to represent the amount (of apples in the basket) with a two-digit number in Base Ten, I would keep adding apples until I had Ten apples int he basket, and if I were to write that number in base 10, I would write, "10 apples. But Base Nine is different because it doesn't have the digit, "9;" in Base Nine, after, "8" is, "10." If you were to write, "Nine Apples" in Base Nine, you would have to write, "I have 10₉ Apples." Notice the little 9 after the 10, this indicates that we're counting in base Nine. But if we were to write that number in Base Ten and specify that we were writing in Base Ten, we would write, "I have 9₁₀ apples."
Now let's count to 19₁₀ apples in Base Nine.
1₉ Apple
2₉ Apple
3₉ Apple
4₉ Apple
5₉ Apple
6₉ Apple
7₉ Apple
8₉ Apple
10₉ Apple (Remember: there''s no 9₉ because in Base Nine, there's 10₁₀ digits including Zero, so we have to add a digit)
11₉ Apple
12₉ Apple
13₉ Apple
14₉ Apple
15₉ Apple
16₉ Apple
17₉ Apple
18₉ Apple
20₉ Apple
21₉ Apple
So if you were to count all of the —occurances of the word, "Apple,"— in Base Ten, you would get 19₁₀, but the numbers to the left go up to 21₉. You could however say that —using the word,"Twenty-One" instead of using the term, "Two One Base Nine"— implies that we're specifying Base Ten instead of Base Nine and that's a valid point. Also we can say that if someone says, "What is nine plus ten?" we can assume that they're expecting an answer in Base Ten since, —ignoring the previous counterpoint and assuming that all of the number that they say are in the same base— implies that they're talking about Base Ten since there's no term, "Nine" in Base Nine. But if we ignore both of those counter points then, Ten₁₀ plus Nine₁₀ equals Nineteen₁₀, which equals Twenty-One₉.
