## Thinking Mathematically (6th Edition)

Multiplication of two given numbers in base sixteen is $\text{71B}{{\text{E}}_{\text{sixteen}}}$.
Since the computation involves base sixteen, the only digit symbols which are allowed are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F. The procedure to multiply two numbers in base sixteenis same as in base ten. $\text{D}{{3}_{\text{sixteen}}}$ $\underline{\times 8{{\text{A}}_{\text{sixteen}}}}$ Hence, first multiply ${{\text{A}}_{\text{sixteen}}}$ in first column from right, with ${{3}_{\text{sixteen}}}$ which is above it in the same column: ${{\text{A}}_{\text{sixteen}}}\times {{3}_{\text{sixteen}}}={{30}_{\text{ten}}}=\left( 1\times 16 \right)+\left( \text{E}\times 1 \right)=1{{\text{E}}_{\text{sixteen}}}$. Now, write ${{\text{E}}_{\text{sixteen}}}$ in the first column from right, below the horizontal line and carry ${{\text{1}}_{\text{sixteen}}}$ to the second column from right: $\overset{1}{\mathop{\text{D}}}\,{{3}_{\text{sixteen}}}$ $\underline{\times 8{{\text{A}}_{\text{sixteen}}}}$ ${{\text{E}}_{\text{sixteen}}}$ Now, multiply ${{\text{A}}_{\text{sixteen}}}$ in first column from right, with ${{\text{D}}_{\text{sixteen}}}$ which is in the second column from right, and add ${{1}_{\text{sixteen}}}$to the product: $\left( {{\text{A}}_{\text{sixteen}}}\times {{\text{D}}_{\text{sixteen}}} \right)+{{1}_{\text{sixteen}}}=130+1={{131}_{\text{ten}}}=\left( 8\times 16 \right)+\left( 3\times 1 \right)={{83}_{\text{sixteen}}}$. Write ${{83}_{\text{sixteen}}}$ in front of ${{\text{E}}_{\text{sixteen}}}$ below the horizontal line: $\text{D}{{3}_{\text{sixteen}}}$ $\underline{\times 8{{\text{A}}_{\text{sixteen}}}}$ $83{{\text{E}}_{\text{sixteen}}}$ Repeat the whole procedure, but with ${{8}_{\text{sixteen}}}$, after placing the symbol $\times$ below ${{\text{E}}_{\text{sixteen}}}$ in $83{{\text{E}}_{\text{sixteen}}}$: $\text{D}{{3}_{\text{sixteen}}}$ $\underline{\times 8{{\text{A}}_{\text{sixteen}}}}$ $83{{\text{E}}_{\text{sixteen}}}$ $\times$ Now, multiply ${{8}_{\text{sixteen}}}$ in second column from right, with ${{3}_{\text{sixteen}}}$ in the first column from right: ${{8}_{\text{sixteen}}}\times {{3}_{\text{sixteen}}}={{24}_{\text{ten}}}=\left( 1\times 16 \right)+\left( 8\times 1 \right)={{18}_{\text{sixteen}}}$. Now, write${{8}_{\text{sixteen}}}$ below ${{3}_{\text{sixteen}}}$in $83{{\text{E}}_{\text{sixteen}}}$, and carry ${{1}_{\text{sixteen}}}$ to the second column from right: $\overset{1}{\mathop{\text{D}}}\,{{3}_{\text{sixteen}}}$ $\underline{\times 8{{\text{A}}_{\text{sixteen}}}}$ $83{{\text{E}}_{\text{sixteen}}}$ $8\times$ Now, multiply ${{8}_{\text{sixteen}}}$ in second column from right, with ${{\text{D}}_{\text{sixteen}}}$ which is in the same column, and add ${{1}_{\text{sixteen}}}$to the product: $\left( {{8}_{\text{sixteen}}}\times {{\text{D}}_{\text{sixteen}}} \right)+{{1}_{\text{sixteen}}}=104+1={{105}_{\text{ten}}}=\left( 6\times 16 \right)+\left( 9\times 1 \right)={{69}_{\text{sixteen}}}$. Now, write ${{69}_{\text{sixteen}}}$ in front of ${{8}_{\text{sixteen}}}$: $\text{D}{{3}_{\text{sixteen}}}$ $\underline{\times 8{{\text{A}}_{\text{sixteen}}}}$ $83{{\text{E}}_{\text{sixteen}}}$ $+\underline{698\times }$ Now, add $83{{\text{E}}_{\text{sixteen}}}$ and ${{698}_{\text{sixteen}}}$ in the manner shown above. First, drop down ${{\text{E}}_{\text{sixteen}}}$ below the symbol $\times$: $\text{D}{{3}_{\text{sixteen}}}$ $\underline{\times 8{{\text{A}}_{\text{sixteen}}}}$ $83{{\text{E}}_{\text{sixteen}}}$ $+\underline{698\times }$ ${{\text{E}}_{\text{sixteen}}}$ Now, ${{3}_{\text{sixteen}}}+{{8}_{\text{sixteen}}}={{11}_{\text{ten}}}=\left( \text{B}\times \text{1} \right)={{\text{B}}_{\text{sixteen}}}$. $\text{D}{{3}_{\text{sixteen}}}$ $\underline{\times 8{{\text{A}}_{\text{sixteen}}}}$ $83{{\text{E}}_{\text{sixteen}}}$ $+\underline{698\times }$ $\text{B}{{\text{E}}_{\text{sixteen}}}$ Now, ${{8}_{\text{sixteen}}}+{{9}_{\text{sixteen}}}={{17}_{\text{ten}}}=\left( 1\times 16 \right)+\left( 1\times \text{1} \right)=\text{1}{{\text{1}}_{\text{sixteen}}}$. $\text{D}{{3}_{\text{sixteen}}}$ $\underline{\times 8{{\text{A}}_{\text{sixteen}}}}$ $83{{\text{E}}_{\text{sixteen}}}$ $+\underline{\overset{1}{\mathop{6}}\,98\times }$ $\text{1B}{{\text{E}}_{\text{sixteen}}}$ Now, ${{1}_{\text{sixteen}}}+{{6}_{\text{sixteen}}}={{7}_{\text{ten}}}=\left( 7\times \text{1} \right)={{7}_{\text{sixteen}}}$: $\text{D}{{3}_{\text{sixteen}}}$ $\underline{\times 8{{\text{A}}_{\text{sixteen}}}}$ $83{{\text{E}}_{\text{sixteen}}}$ $+\underline{698\times }$ $\text{71B}{{\text{E}}_{\text{sixteen}}}$ Now, to check whether the above obtained solution is correct, perform the multiplication by converting each number to base ten: $\text{D}{{3}_{\text{sixteen}}}=211$, $8{{\text{A}}_{\text{sixteen}}}=138$and $\text{71B}{{\text{E}}_{\text{sixteen}}}=29118$. Since, $211\times 138$ indeed equals 29118, the solution obtained is correct. Hence, multiplication of two given numbers in base sixteen is $\text{71B}{{\text{E}}_{\text{sixteen}}}$.