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Answer
Success!
We are done. 117 converted from decimal to binary notation equals 11101012.

↓Steps Explained:↓

Convert 117 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 >= 117

20 = 1

21 = 2

22 = 4

23 = 8

24 = 16

25 = 32

26 = 64

27 = 128 <--- Stop: This is greater than 117

Since 128 is greater than 117, 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 117 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 <= 117, 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 117 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 <= 117, 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 117 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 <= 117, 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 117 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 > 117, so we assign a 0 for this digit.

Our total sum remains the same at 112

Our binary notation is now equal to 1110


22 = 4

The highest coefficient less than 1 we can multiply this by to stay under 117 is 1

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

Add our new value to our running total, we get:
112 + 4 = 116

This is <= 117, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 116

Our binary notation is now equal to 11101


21 = 2

The highest coefficient less than 1 we can multiply this by to stay under 117 is 1

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

Add our new value to our running total, we get:
116 + 2 = 118

This is > 117, so we assign a 0 for this digit.

Our total sum remains the same at 116

Our binary notation is now equal to 111010


20 = 1

The highest coefficient less than 1 we can multiply this by to stay under 117 is 1

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

Add our new value to our running total, we get:
116 + 1 = 117

This = 117, so we assign our outside coefficient of 1 for this digit.

Our new sum becomes 117

Our binary notation is now equal to 1110101


Final Answer

We are done. 117 converted from decimal to binary notation equals 11101012.
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