The binary systems is a way to encode numbers. We are used to the [[Decimal System]] using the numerals $0-9$. In the binary system, we are using only two numerals $0$ and $1$. These [[Bit|bits]] represent numbers in binary. This is the $2$ system, i.e. a representation of base $2$ with two digits: $0$ and $1$. As an example, we will again use the number $158_{10}$. The index $10$ here means that the number is represented in the [[Decimal System|decimal system]]. The table for the binary system uses the same representation as above. Instead of in the note of the [[Decimal System|decimal system]] . Instead of decomposing a number into powers of $10$, we now use powers[^1] of $2$. ![[binary_system.excalidraw.light.svg]] As a formula, this means $158=1\cdot 128 + 1\cdot 16 + 1 \cdot 8 + 1\cdot 4 +1\cdot 2$. In summary, $10011110_2=158_{10}$. >[!read]- Further Reading >- [[Decimal System]] >- [[Hexadecimal System]] >- [[Modular Exponentiation]] >[!ref]- References [^1]: A power in math is a shorthand notation. $x^3$ (speak "$x$ to the power of $3$) means that we multiply $x$ three times with itself: $x^3 = x\cdot x\cdot x$.