# What does $23_4$ mean?

I just saw this on a mathematical clock for $11$, i.e $23_4=11$:

$\qquad \qquad \qquad \qquad \qquad$ I guess it is some notation from algebra. But since algebra was never my favorite field of maths, I don't know this notation. Any explanations are welcome ;-))! Thanks

• 11 in base 10 is same as 23 in base 4 : $$(23)_4 = (11)_{10}$$ – ganeshie8 Oct 4 '14 at 10:46
• It is an interesting question and the clock is also meaningful.+1 – Paul Oct 4 '14 at 11:07

This denotes the number $11$ in base $4$. In everyday life, we write our numbers in base $10$.

$23_4$ is to be read as: $$2\cdot 4 + 3.$$ In general, $$(a_n...a_0) _ g = \sum_{i=0}^n a_i g^i = a_n g^n + a_{n-1}g^{n-1} + ... + a_1 g + a_0,$$ where the $a_i$ are chosen to lie in $\{0,...,g-1\}$.

EDIT: I have edited this post to write $2\cdot 4 +3$ rather than $3+2\cdot 4$. However, I still think that it is easier to decipher a (long) number such as $(2010221021)_3$ from right to left, simply by increasing the powers of $3$, rather than first checking that the highest occuring power of $3$ is $3^9$ and then going from left to right.

• Not reversing the order will make it more readable. – Yves Daoust Oct 4 '14 at 10:55
• Personally, I prefer it this way, because I can simply increase the power of $g$ as I go from right to left, whereas I first have to check which the highest occuring power of $g$ is when I go from left to right. Nonetheless, I read base 10 numbers from left to right, but that is probably due to the fact that we're used to thinking of numbers in base 10 and writing them this way... – Oliver Braun Oct 4 '14 at 11:02
• If you write $23_4$, then $2\cdot4+3$ is more readable than $3+2\cdot4$. I also use to write polynomials by decreasing degrees, but this is just a matter of taste. – Yves Daoust Oct 4 '14 at 11:15
• That's a matter of opinion, is it not? – Oliver Braun Oct 4 '14 at 11:17
• Not at all. In one case there is an inversion and not in the other. The writing order is a matter of convention and habit and I have no objection when they are used in isolation, but when you bring the expressions together, then it starts to matter. – Yves Daoust Oct 4 '14 at 11:19

$4$ is the base, $(23)_4$ means $2 \cdot 4^1 + 3 \cdot 4^0 = 11$. Similarly, if $10$ is the base, then $(23)_{10}$ means $2 \cdot 10^1 + 3 \cdot 10^0 = 23$.

• That is the notation used by Knuth. – mvw Dec 20 '14 at 11:45

The base or radix of a number denotes how many unique digits are used by the numeral system that is representing the given number. Usually the radix is written as a subscript, as in your example $23_4$, where the radix is $4$.

Also note that when the radix is omitted, it is usually assumed to be $10$. So the number $23_4$ is equal to $11_{10}$ or simply $11$. Here is how we can convert $23_4$ to $11$ $$23_4=\left(2\cdot 4^1\right)+\left(3\cdot 4^0\right) =8+3=11$$

In general, $$\left(\alpha_{n-1}\dots\alpha_1\alpha_0\right)_{\beta}$$ $$= \left(\alpha_{n-1}\cdot\beta^{n-1}\right)+\cdots+ \left(\alpha_1\cdot\beta^1\right)+ \left(\alpha_0\cdot\beta^0\right)$$