Convert 39 from decimal to binary (base 2) notation:

Start by raising our base of 2 to a power starting at 0 and increasing by 1 until it is >= 39

2

2

2

2

2

2

2

Since 64 is greater than 39, we use 1 power less as our starting point which equals 5.

Now start building our binary notation working backwards from a power of 5.

We start with a total sum of 0:

2

The highest coefficient less than 1 we can multiply this by to stay under 39 is 1.

Multiplying this coefficient by our original value, we get: 1 * 32 = 32.

Adding our new value to our running total, we get: 0 + 32 = 32.

This is <= 39, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 32.

Our binary notation is now equal to 1

2

The highest coefficient less than 1 we can multiply this by to stay under 39 is 1.

Multiplying this coefficient by our original value, we get: 1 * 16 = 16.

Adding our new value to our running total, we get: 32 + 16 = 48.

This is > 39, so we assign a 0 for this digit.

Our total sum remains the same at 32.

Our binary notation is now equal to 10

2

The highest coefficient less than 1 we can multiply this by to stay under 39 is 1.

Multiplying this coefficient by our original value, we get: 1 * 8 = 8.

Adding our new value to our running total, we get: 32 + 8 = 40.

This is > 39, so we assign a 0 for this digit.

Our total sum remains the same at 32.

Our binary notation is now equal to 100

2

The highest coefficient less than 1 we can multiply this by to stay under 39 is 1.

Multiplying this coefficient by our original value, we get: 1 * 4 = 4.

Adding our new value to our running total, we get: 32 + 4 = 36.

This is <= 39, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 36.

Our binary notation is now equal to 1001

2

The highest coefficient less than 1 we can multiply this by to stay under 39 is 1.

Multiplying this coefficient by our original value, we get: 1 * 2 = 2.

Adding our new value to our running total, we get: 36 + 2 = 38.

This is <= 39, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 38.

Our binary notation is now equal to 10011

2

The highest coefficient less than 1 we can multiply this by to stay under 39 is 1.

Multiplying this coefficient by our original value, we get: 1 * 1 = 1.

Adding our new value to our running total, we get: 38 + 1 = 39.

This = 39, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 39.

Our binary notation is now equal to 100111

We are done. 39 converted from decimal to binary notation equals