The decimal system is all around us: we use the numbers $0-9$ and combinations of them to express quantities ranging the number of breads in a basket to the price of a stock. We rarely think about that this system is arbitrary: nobody dictates that we have to use 10 numerals ($0-9$) instead of say $16$ ([[Hexadecimal System|hexadecimal system]]) or $2$ ([[binary system]]). To better understand how this works, let's first look at the decimal system. Numbers in the decimal system (i.e. the ordinary numbers) are represented using powers of $10$. The power is a shorthand notation for multiplication: $x^3=x\cdot x\cdot x$. Here $x$ is the base and $3$ is the exponent. The power with exponent $0$ is defined as $x^0=1$. This is where the name decimal system comes from. It is the $10$ system because it allows exactly $10$ digits: $0-9$. If we want to represent a number larger than $9$, we have to start at a new place: after $9$ comes $10$ (read $1$-**pause**-$0$). We can express the number $158$ as a power of $10$ by writing it as $158=1\cdot 100 + 5 \cdot 10 + 8 \cdot 1$. The prefactors are the different digits of the number. The factors $1$, $10$ and $100$ are powers of 10: $1=10^0$, $10=10^1$, $100=10^2$. For larger numbers, we need larger powers of $10$. The representation in a table is a little clearer. ![[decimal_system.excalidraw.light.svg]] The first line represents the powers of 10. The second line contains the actual number, i.e. the prefactors of the individual powers. >[!read]- Further Reading >- [[Binary System]] >- [[Hexadecimal System]] >[!ref]- References