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Convert 164 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 >= 164

2⁰ = 1

2¹ = 2

2² = 4

2³ = 8

2⁴ = 16

2⁵ = 32

2⁶ = 64

2⁷ = 128

2⁸ = 256 <--- Stop: This is greater than 164

Since 256 is greater than 164, 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 164 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 <= 164, 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 164 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 > 164, 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 164 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 <= 164, 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 164 is 1

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

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

This is > 164, so we assign a 0 for this digit.

Our total sum remains the same at 160

Our binary notation is now equal to 1010

2³ = 8

The highest coefficient less than 1 we can multiply this by to stay under 164 is 1

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

Add our new value to our running total, we get:
160 + 8 = 168

This is > 164, so we assign a 0 for this digit.

Our total sum remains the same at 160

Our binary notation is now equal to 10100

2² = 4

The highest coefficient less than 1 we can multiply this by to stay under 164 is 1

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

Add our new value to our running total, we get:
160 + 4 = 164

This = 164, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 164

Our binary notation is now equal to 101001

2¹ = 2

The highest coefficient less than 1 we can multiply this by to stay under 164 is 1

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

Add our new value to our running total, we get:
164 + 2 = 166

This is > 164, so we assign a 0 for this digit.

Our total sum remains the same at 164

Our binary notation is now equal to 1010010

2⁰ = 1

The highest coefficient less than 1 we can multiply this by to stay under 164 is 1

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

Add our new value to our running total, we get:
164 + 1 = 165

This is > 164, so we assign a 0 for this digit.

Our total sum remains the same at 164

Our binary notation is now equal to 10100100

Final Answer