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Convert 257 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 >= 257

2⁰ = 1

2¹ = 2

2² = 4

2³ = 8

2⁴ = 16

2⁵ = 32

2⁶ = 64

2⁷ = 128

2⁸ = 256

2⁹ = 512 <--- Stop: This is greater than 257

Since 512 is greater than 257, we use 1 power less as our starting point which equals 8

Build binary notation

Work 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 257 is 1

Multiplying this coefficient by our original value, we get: 1 * 256 = 256

Add our new value to our running total, we get:
0 + 256 = 256

This is <= 257, 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 257 is 1

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

Add our new value to our running total, we get:
256 + 128 = 384

This is > 257, 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 257 is 1

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

Add our new value to our running total, we get:
256 + 64 = 320

This is > 257, 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 257 is 1

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

Add our new value to our running total, we get:
256 + 32 = 288

This is > 257, 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 257 is 1

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

Add our new value to our running total, we get:
256 + 16 = 272

This is > 257, 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 257 is 1

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

Add our new value to our running total, we get:
256 + 8 = 264

This is > 257, 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 257 is 1

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

Add our new value to our running total, we get:
256 + 4 = 260

This is > 257, 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 257 is 1

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

Add our new value to our running total, we get:
256 + 2 = 258

This is > 257, 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 257 is 1

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

Add our new value to our running total, we get:
256 + 1 = 257

This = 257, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 257

Our binary notation is now equal to 100000001

Final Answer