Decimal and Binary Number Systems

The decimal (base-10) system. The number system used in everyday arithmetic is the DECIMAL system, built on TEN distinct digit symbols ($0$ through $9$) and place values that are successive powers of ten: reading from the rightmost (units) digit outward, the place values are $10^0=1$, $10^1=10$, $10^2=100$, $10^3=1000$, and so on. A decimal number's value is simply the sum of each digit multiplied by its own place value -- e.g. $247 = 2\times10^2 + 4\times10^1 + 7\times10^0$.

The binary (base-2) system. The BINARY system uses only TWO digit symbols, called BITS (binary digits): $0$ and $1$, with place values that are successive powers of TWO instead of ten: $2^0=1$, $2^1=2$, $2^2=4$, $2^3=8$, $2^4=16$, and so on. A binary number's decimal value is found the same way as for decimal -- summing each bit multiplied by its own place value -- e.g. the binary number $1011_2$ has, reading from the left, bits $1,0,1,1$ occupying place values $2^3,2^2,2^1,2^0$, so

Converting decimal to binary. Converting a decimal number into its binary equivalent is done by REPEATED DIVISION by $2$, recording the remainder ($0$ or $1$) at each step, and reading the remainders in REVERSE order (last remainder first). Dividing the number by $2$ repeatedly, and keeping track of the remainder at each division, exhausts every place value from lowest to highest, so reading the remainders bottom-to-top reconstructs the binary number from its highest bit down to its lowest. …