3 What is 162?

Without any special indicator, 162 is a base-10 (decimal) number. Of course, we all know the value of one hundred and sixty two, don't we?

Afterall,

$$162 = 100 + 60 + 2$$ (1)

$$= 1\times 100 + 6\times 10 + 2 \times 1$$ (2)

$$= 1\times 10^2 + 6\times 10^1 + 2\times 10^0$$ (3)

There is nothing to it! Note that in the last line, we use a sum of multiples of powers of 10.

What if we change our base to 8? In other words, what is the value represented by $162_8$? By the way, it is a convention to use a subscript to indicate the base of a number. If you use a regular text editor, you can also use parantheses to surround the base, like 162(8), or use the C notation 0162.

Getting back to our example, $162_8$ can be broken down as follows:

$$162_8 = 100_8 + 60_8 + 2_8$$ (4)

$$= 1\times 100_8 + 6 \times 10_8 + 2\times 1_8$$ (5)

$$= 1\times 8^2 + 6\times 8^1 + 2\times 8^0$$ (6)

$$= 64 + 48 + 2$$ (7)

$$= 114$$ (8)

Now, isn't that cool? As we change the base of a number, the value it represent changes! How about the value of $162_4$?

That's a trick question! There is no such number because the digit 6 is greater than or equal to the base (4). This is hardly surprising. After all, there is no single decimal to represent the value of ten.