Base Change Conversions Calculator

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Convert 21 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 >= 21

2⁰ = 1

2¹ = 2

2² = 4

2³ = 8

2⁴ = 16

2⁵ = 32 <--- Stop: This is greater than 21

Since 32 is greater than 21, we use 1 power less as our starting point which equals 4

Build binary notation

Work backwards from a power of 4

We start with a total sum of 0:

2⁴ = 16

The highest coefficient less than 1 we can multiply this by to stay under 21 is 1

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

Add our new value to our running total, we get:
0 + 16 = 16

This is <= 21, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 16

Our binary notation is now equal to 1

2³ = 8

The highest coefficient less than 1 we can multiply this by to stay under 21 is 1

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

Add our new value to our running total, we get:
16 + 8 = 24

This is > 21, so we assign a 0 for this digit.

Our total sum remains the same at 16

Our binary notation is now equal to 10

2² = 4

The highest coefficient less than 1 we can multiply this by to stay under 21 is 1

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

Add our new value to our running total, we get:
16 + 4 = 20

This is <= 21, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 20

Our binary notation is now equal to 101

2¹ = 2

The highest coefficient less than 1 we can multiply this by to stay under 21 is 1

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

Add our new value to our running total, we get:
20 + 2 = 22

This is > 21, so we assign a 0 for this digit.

Our total sum remains the same at 20

Our binary notation is now equal to 1010

2⁰ = 1

The highest coefficient less than 1 we can multiply this by to stay under 21 is 1

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

Add our new value to our running total, we get:
20 + 1 = 21

This = 21, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 21

Our binary notation is now equal to 10101

Final Answer