Convert 191 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 >= 191

2

2

2

2

2

2

2

2

2

Since 256 is greater than 191, we use 1 power less as our starting point which equals 7.

Now start building our binary notation working backwards from a power of 7.

We start with a total sum of 0:

2

The highest coefficient less than 1 we can multiply this by to stay under 191 is 1.

Multiplying this coefficient by our original value, we get: 1 * 128 = 128.

Adding our new value to our running total, we get: 0 + 128 = 128.

This is <= 191, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 128.

Our binary notation is now equal to 1

2

The highest coefficient less than 1 we can multiply this by to stay under 191 is 1.

Multiplying this coefficient by our original value, we get: 1 * 64 = 64.

Adding our new value to our running total, we get: 128 + 64 = 192.

This is > 191, so we assign a 0 for this digit.

Our total sum remains the same at 128.

Our binary notation is now equal to 10

2

The highest coefficient less than 1 we can multiply this by to stay under 191 is 1.

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

Adding our new value to our running total, we get: 128 + 32 = 160.

This is <= 191, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 160.

Our binary notation is now equal to 101

2

The highest coefficient less than 1 we can multiply this by to stay under 191 is 1.

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

Adding our new value to our running total, we get: 160 + 16 = 176.

This is <= 191, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 176.

Our binary notation is now equal to 1011

2

The highest coefficient less than 1 we can multiply this by to stay under 191 is 1.

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

Adding our new value to our running total, we get: 176 + 8 = 184.

This is <= 191, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 184.

Our binary notation is now equal to 10111

2

The highest coefficient less than 1 we can multiply this by to stay under 191 is 1.

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

Adding our new value to our running total, we get: 184 + 4 = 188.

This is <= 191, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 188.

Our binary notation is now equal to 101111

2

The highest coefficient less than 1 we can multiply this by to stay under 191 is 1.

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

Adding our new value to our running total, we get: 188 + 2 = 190.

This is <= 191, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 190.

Our binary notation is now equal to 1011111

2

The highest coefficient less than 1 we can multiply this by to stay under 191 is 1.

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

Adding our new value to our running total, we get: 190 + 1 = 191.

This = 191, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 191.

Our binary notation is now equal to 10111111

We are done. 191 converted from decimal to binary notation equals