1+1 no longer equals to 2

Number Theory Level 2

1 0 1 1 0 1 0 + 1 1 0 0 1 1 \large{\begin{array}{cccccccc} &&1 &0 & 1 & 1& 0 & 1&0\\ +&&& 1 & 1 & 0& 0 & 1&1\\ \hline && & & & & & &\\ && & & & & & & \\ && & & & & & & \\ \hline \end{array}}

Considering the addition of binary numbers above, what is the value of this sum written in the decimal base?


The answer is 141.

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Worranat Pakornrat by Worranat Pakornrat , 36, Thailand

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1 solution

In binary base system, only digits 1 & 0 can be used with the value doubling in the next larger unit. That means, in this case, 1+1 = 10 ( 1 0 2 10_{2} = 2 10 2_{10} ).

Then in the addition above, 1+0 = 1 as in decimal process, but when it's 1+1 = 10, the digit 1 (red) will be added up to the next larger unit, leaving 0 down as shown.

As a result, 101101 0 2 1011010_{2} + 11001 1 2 110011_{2} = 1000110 1 2 10001101_{2}

Now converting the binary number to decimal one by summation of their unit values, we will get:

1000110 1 2 10001101_{2} = 2 7 2^7 + 2 3 2^3 + 2 2 2^2 + 2 0 2^0 = 141.

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