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Convert 109 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 >= 109

2⁰ = 1

2¹ = 2

2² = 4

2³ = 8

2⁴ = 16

2⁵ = 32

2⁶ = 64

2⁷ = 128 <--- Stop: This is greater than 109

Since 128 is greater than 109, 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 109 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 <= 109, 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 109 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 <= 109, 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 109 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 > 109, so we assign a 0 for this digit.

Our total sum remains the same at 96

Our binary notation is now equal to 110

2³ = 8

The highest coefficient less than 1 we can multiply this by to stay under 109 is 1

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

Add our new value to our running total, we get:
96 + 8 = 104

This is <= 109, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 104

Our binary notation is now equal to 1101

2² = 4

The highest coefficient less than 1 we can multiply this by to stay under 109 is 1

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

Add our new value to our running total, we get:
104 + 4 = 108

This is <= 109, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 108

Our binary notation is now equal to 11011

2¹ = 2

The highest coefficient less than 1 we can multiply this by to stay under 109 is 1

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

Add our new value to our running total, we get:
108 + 2 = 110

This is > 109, so we assign a 0 for this digit.

Our total sum remains the same at 108

Our binary notation is now equal to 110110

2⁰ = 1

The highest coefficient less than 1 we can multiply this by to stay under 109 is 1

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

Add our new value to our running total, we get:
108 + 1 = 109

This = 109, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 109

Our binary notation is now equal to 1101101

Final Answer