**⌚ September 2, 2014**

If we can convert from decimal (base 10) to binary (base 2), is it possible to to convert in the other direction from binary to decimal?

Yes, it is, and we will see that this is a much easier conversion to perform since it only involves addition.

## Convert Binary 11000111 to Decimal

First, create a binary place value chart containing as many places as there are digits in the binary number to convert. The number 11000111 contains eight digits, so we create a binary place value chart containing eight places.

Right-align the number with the place value chart.

To convert, start from the left-most digit and sum every place that contains a 1. Ignore all places that contain a 0.

The decimal value of 11000111 is 199.

It really does not matter which side to begin adding from, but beginning at the most significant place value allows us to estimate how large the answer will be before we are finished. If we start from the least significant digit, 1, then we have no idea of the value until complete. Beginning from the left tells us that the answer will be at least 128. We can begin with a close approximation and then refine its accuracy as we sum to the right.

Why does this work? View the chart below. We are actually multiplying the value of each place the specified number of times (indicated by the digit it occupies) and then summing all results just as we do with decimal numbers. Since anything multiplied by zero is zero, we can ignore any place containing a zero.

## Convert Binary 100001000 to Decimal

This is nine digits in length, so create a binary place value chart with nine places.

Right-align 100001000 with the place value chart.

Sum the value of all places containing a 1 digit. Ignore the places with zeros.

The decimal value of 100001000 is 264.

## Practice

Proficiency improves through practice, so convert the following binary values into their decimal values. Use pencil and paper. *Do not use a calculator since that will hinder learning at this point*. The key is to create a correct place value chart. Once that is complete, the rest is simple addition. See Lesson 2 for a review in creating a binary place value chart. As you add from left to right, keep a running total in your mind.

- 11000111
- 1001
- 110011
- 10101010
- 11000000
- 1010
- 10101000
- 1000000001
- 1111101000
- 1111110011111010

### Answers (Try not to peek!)

- 199
- 9
- 51
- 170
- 192
- 10 (Looks like binary, doesn’t it?)
- 168
- 513
- 1,000 (Another binary “look-alike”)
- 64,762