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Convert 159 from decimal to binary

(base 2) notation:

Power Test

Raise our base of 2 to a power

Start at 0 and increasing by 1 until it is >= 159

20 = 1

21 = 2

22 = 4

23 = 8

24 = 16

25 = 32

26 = 64

27 = 128

28 = 256 <--- Stop: This is greater than 159

Since 256 is greater than 159, we use 1 power less as our starting point which equals 7

Build binary notation

Work 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 159 is 1

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

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

This is <= 159, 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 159 is 1

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

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

This is > 159, 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 159 is 1

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

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

This is > 159, so we assign a 0 for this digit.

Our total sum remains the same at 128

Our binary notation is now equal to 100


24 = 16

The highest coefficient less than 1 we can multiply this by to stay under 159 is 1

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

Add our new value to our running total, we get:
128 + 16 = 144

This is <= 159, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 144

Our binary notation is now equal to 1001


23 = 8

The highest coefficient less than 1 we can multiply this by to stay under 159 is 1

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

Add our new value to our running total, we get:
144 + 8 = 152

This is <= 159, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 152

Our binary notation is now equal to 10011


22 = 4

The highest coefficient less than 1 we can multiply this by to stay under 159 is 1

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

Add our new value to our running total, we get:
152 + 4 = 156

This is <= 159, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 156

Our binary notation is now equal to 100111


21 = 2

The highest coefficient less than 1 we can multiply this by to stay under 159 is 1

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

Add our new value to our running total, we get:
156 + 2 = 158

This is <= 159, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 158

Our binary notation is now equal to 1001111


20 = 1

The highest coefficient less than 1 we can multiply this by to stay under 159 is 1

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

Add our new value to our running total, we get:
158 + 1 = 159

This = 159, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 159

Our binary notation is now equal to 10011111


Final Answer


We are done. 159 converted from decimal to binary notation equals 100111112.