Problem
 Express the decimal numbers 19 and 10 as binary numbers.
 Add the binary numbers together and convert the sum to a decimal number.
 Multiply the two binary numbers and convert the product to decimal form.
Background
Mathematical operations are carried out in binary arithmetic in the same way as in decimal arithmetic. As an example of the similarity between decimal and binary arithmetic, when we add 75 to 129, we add 9 to 5 in the righthand column and get 4 plus 1 to be carried to the next column. In binary addition, we have only the digits 1 and 0; when we add 1 to 1 we get 0 plus 1 to carry to the next column. In subtraction, when we subtract 1 from 0 we must borrow 1 from the next nonzero column to the left. Multiplication is identical in the two systems. To express a decimal number as a binary number, we subtract the largest power of 2 that is less than the decimal number, then follow the same procedure with the remainders until we end up with 1 or 0.
Solution
 $19=2^{4}+2^{1}+2^{0}=10011;10=2^{3}+2^{1}=1010.$
 $10011+1010=11101=2^{4}+2^{3}+2^{2}+2^{0}=16+8+4+1=29.$
 $10011\times 1010=$
${\begin{array}{c}10\,011\\1\,010\\\hline 100\,110\\10\,011\,0\\\hline 10\,111\,110=2^{7}+2^{5}+2^{4}+2^{3}+2^{2}+2^{1}+0\\\qquad =128+32+16+8+4+2\\=190.\end{array}}$
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