 # Base Change Conversions Calculator

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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
20 = 1
21 = 2
22 = 4
23 = 8
24 = 16
25 = 32
26 = 64
27 = 128
28 = 256 <--- Stop: This is greater than 191

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:

27 = 128.
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

26 = 64.
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

25 = 32.
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

24 = 16.
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

23 = 8.
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

22 = 4.
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

21 = 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

20 = 1.
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 101111112.