# Day 10: Binary Numbers

# Day 10: Binary Numbers

Terms you'll find helpful in completing today's challenge are outlined below, along with sample Java code (where appropriate).

## Radix (Base)

The number of digits that can be used to represent a number in a positional number system. The decimal number system (base-) has digits (); the binary (base-) number system has digits ().

We think in terms of base-, because the decimal number system is the only one many people need in everyday life. For situations where there is a need to specify a number's radix, number having radix should be written as .

## Binary to Decimal Conversion

In the same way that , a binary number having digits in the form of can be converted to decimal by summing the result for each where , is the most significant bit, and is the least significant bit.

For example: is evaluated as

## Decimal to Binary Conversion

To convert an integer from decimal to binary, repeatedly divide your base- number, , by . The dividend at each step should be the result of the integer division at each step . The remainder at each step of division is a single digit of the binary equivalent of ; if you then read each remainder in order from the last remainder to the first (demonstrated below), you have the entire binary number.

For example: . After performing the steps outlined in the above paragraph, the remainders form (the binary equivalent of ) when read from the bottom up:

This can be expressed in pseudocode as:

```
while(n > 0):
remainder = n%2;
n = n/2;
Insert remainder to front of a list or push onto a stack
Print list or stack
```

Many languages have built-in functions for converting numbers from decimal to binary. To convert an integer, , from decimal to a String of binary numbers in Java, you can use the *Integer.toBinaryString(n)* function.

**Note:** The algorithm discussed here is for converting integers; converting fractional numbers is a similar (but different) process.