Base Change Conversions Calculator

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Convert 127 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 >= 127

2⁰ = 1

2¹ = 2

2² = 4

2³ = 8

2⁴ = 16

2⁵ = 32

2⁶ = 64

2⁷ = 128 <--- Stop: This is greater than 127

Since 128 is greater than 127, we use 1 power less as our starting point which equals 6

Build binary notation

Work backwards from a power of 6

We start with a total sum of 0:

2⁶ = 64

The highest coefficient less than 1 we can multiply this by to stay under 127 is 1

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

Add our new value to our running total, we get:
0 + 64 = 64

This is <= 127, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 64

Our binary notation is now equal to 1

2⁵ = 32

The highest coefficient less than 1 we can multiply this by to stay under 127 is 1

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

Add our new value to our running total, we get:
64 + 32 = 96

This is <= 127, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 96

Our binary notation is now equal to 11

2⁴ = 16

The highest coefficient less than 1 we can multiply this by to stay under 127 is 1

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

Add our new value to our running total, we get:
96 + 16 = 112

This is <= 127, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 112

Our binary notation is now equal to 111

2³ = 8

The highest coefficient less than 1 we can multiply this by to stay under 127 is 1

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

Add our new value to our running total, we get:
112 + 8 = 120

This is <= 127, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 120

Our binary notation is now equal to 1111

2² = 4

The highest coefficient less than 1 we can multiply this by to stay under 127 is 1

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

Add our new value to our running total, we get:
120 + 4 = 124

This is <= 127, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 124

Our binary notation is now equal to 11111

2¹ = 2

The highest coefficient less than 1 we can multiply this by to stay under 127 is 1

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

Add our new value to our running total, we get:
124 + 2 = 126

This is <= 127, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 126

Our binary notation is now equal to 111111

2⁰ = 1

The highest coefficient less than 1 we can multiply this by to stay under 127 is 1

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

Add our new value to our running total, we get:
126 + 1 = 127

This = 127, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 127

Our binary notation is now equal to 1111111

Final Answer