# Binary Number System

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In a binary number (or base two) system, we use only two digits: 0 and 1. The place value for a base two number is shown as follows:

 Place Value Sixty-fours Thirty-twos Sixteens Eights Fours Twos Units/ Ones 26 25 24 23 22 21 20

## Convert From Binary To Decimal

Like base ten numbers, we can determine the value of a base two number by placing each digit in its respective place value, then writing its expanded notation.

Example:
Find the value of each of the following binary numbers, giving the values in decimal numbers:
a)         1012                 b)         10112               c)         101012

Solution:
a)

 24 23 22 21 20 1 0 1

1012     = 1 × 22 + 0 × 21 + 1 × 20
= 4 + 0 + 1
= 510

b)

 24 23 22 21 20 1 0 1 1

10112   = 1 × 23 + 0 × 22 + 1 × 21 + 1 × 20
= 8 + 0 + 2 + 1
= 1110

c)

 24 23 22 21 20 1 0 1 0 1

101012   = 1 × 24 + 0 × 23 + 1 × 22 + 0 × 21 + 1 × 20
= 16 + 0 + 4 + 0 + 1
= 2110

## Convert From Decimal To Binary

We can convert a decimal number into a binary number by repeatedly dividing the base ten number by two. Then, write the remainders from the bottom to the top as the answer:

Example:
Write each of the following base ten numbers as a binary number:
a)         510                                b)         7810

Solution:

a)

510  = 1012

b)

7810      = 10011102

Base 10 to Base 2 (Decimal to Binary) Conversion
A demonstration of the repeated divide by 2 method for converting numbers from base 10 (or decimal) into base 2 (or binary) form.

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