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Convert 256 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 >= 256
20 = 1
21 = 2
22 = 4
23 = 8
24 = 16
25 = 32
26 = 64
27 = 128
28 = 256 <--- Stop: This is equal to 256

Since 256 is equal to 256, we use our current power as our starting point which equals 8.

Now start building our binary notation working backwards from a power of 8.
We start with a total sum of 0:

28 = 256.
The highest coefficient less than 1 we can multiply this by to stay under 256 is 1.
Multiplying this coefficient by our original value, we get: 1 * 256 = 256.
Adding our new value to our running total, we get: 0 + 256 = 256.

This = 256, so we assign our outside coefficient of 1 for this digit.
Our new sum becomes 256.
Our binary notation is now equal to 1

27 = 128.
The highest coefficient less than 1 we can multiply this by to stay under 256 is 1.
Multiplying this coefficient by our original value, we get: 1 * 128 = 128.
Adding our new value to our running total, we get: 256 + 128 = 384.

This is > 256, so we assign a 0 for this digit.
Our total sum remains the same at 256.
Our binary notation is now equal to 10

26 = 64.
The highest coefficient less than 1 we can multiply this by to stay under 256 is 1.
Multiplying this coefficient by our original value, we get: 1 * 64 = 64.
Adding our new value to our running total, we get: 256 + 64 = 320.

This is > 256, so we assign a 0 for this digit.
Our total sum remains the same at 256.
Our binary notation is now equal to 100

25 = 32.
The highest coefficient less than 1 we can multiply this by to stay under 256 is 1.
Multiplying this coefficient by our original value, we get: 1 * 32 = 32.
Adding our new value to our running total, we get: 256 + 32 = 288.

This is > 256, so we assign a 0 for this digit.
Our total sum remains the same at 256.
Our binary notation is now equal to 1000

24 = 16.
The highest coefficient less than 1 we can multiply this by to stay under 256 is 1.
Multiplying this coefficient by our original value, we get: 1 * 16 = 16.
Adding our new value to our running total, we get: 256 + 16 = 272.

This is > 256, so we assign a 0 for this digit.
Our total sum remains the same at 256.
Our binary notation is now equal to 10000

23 = 8.
The highest coefficient less than 1 we can multiply this by to stay under 256 is 1.
Multiplying this coefficient by our original value, we get: 1 * 8 = 8.
Adding our new value to our running total, we get: 256 + 8 = 264.

This is > 256, so we assign a 0 for this digit.
Our total sum remains the same at 256.
Our binary notation is now equal to 100000

22 = 4.
The highest coefficient less than 1 we can multiply this by to stay under 256 is 1.
Multiplying this coefficient by our original value, we get: 1 * 4 = 4.
Adding our new value to our running total, we get: 256 + 4 = 260.

This is > 256, so we assign a 0 for this digit.
Our total sum remains the same at 256.
Our binary notation is now equal to 1000000

21 = 2.
The highest coefficient less than 1 we can multiply this by to stay under 256 is 1.
Multiplying this coefficient by our original value, we get: 1 * 2 = 2.
Adding our new value to our running total, we get: 256 + 2 = 258.

This is > 256, so we assign a 0 for this digit.
Our total sum remains the same at 256.
Our binary notation is now equal to 10000000

20 = 1.
The highest coefficient less than 1 we can multiply this by to stay under 256 is 1.
Multiplying this coefficient by our original value, we get: 1 * 1 = 1.
Adding our new value to our running total, we get: 256 + 1 = 257.

This is > 256, so we assign a 0 for this digit.
Our total sum remains the same at 256.
Our binary notation is now equal to 100000000

We are done. 256 converted from decimal to binary notation equals 1000000002.