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Convert 47 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 >= 47

2⁰ = 1

2¹ = 2

2² = 4

2³ = 8

2⁴ = 16

2⁵ = 32

2⁶ = 64 <--- Stop: This is greater than 47

Since 64 is greater than 47, we use 1 power less as our starting point which equals 5

Build binary notation

Work backwards from a power of 5

We start with a total sum of 0:

2⁵ = 32

The highest coefficient less than 1 we can multiply this by to stay under 47 is 1

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

Add our new value to our running total, we get:
0 + 32 = 32

This is <= 47, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 32

Our binary notation is now equal to 1

2⁴ = 16

The highest coefficient less than 1 we can multiply this by to stay under 47 is 1

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

Add our new value to our running total, we get:
32 + 16 = 48

This is > 47, so we assign a 0 for this digit.

Our total sum remains the same at 32

Our binary notation is now equal to 10

2³ = 8

The highest coefficient less than 1 we can multiply this by to stay under 47 is 1

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

Add our new value to our running total, we get:
32 + 8 = 40

This is <= 47, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 40

Our binary notation is now equal to 101

2² = 4

The highest coefficient less than 1 we can multiply this by to stay under 47 is 1

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

Add our new value to our running total, we get:
40 + 4 = 44

This is <= 47, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 44

Our binary notation is now equal to 1011

2¹ = 2

The highest coefficient less than 1 we can multiply this by to stay under 47 is 1

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

Add our new value to our running total, we get:
44 + 2 = 46

This is <= 47, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 46

Our binary notation is now equal to 10111

2⁰ = 1

The highest coefficient less than 1 we can multiply this by to stay under 47 is 1

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

Add our new value to our running total, we get:
46 + 1 = 47

This = 47, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 47

Our binary notation is now equal to 101111

Final Answer