0.4: Egyptian Multiplication and Division (optional)
- Page ID
- 25306
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Duplation is a multiplication strategy that students can learn to help further develop their number sense and ability to multiply numbers. Duplation or also referred to as mediation can be dated back to being one of the earliest records of multiplication used by the Egyptians (Ancient Multiplication Methods, n.d. para.1). The Duplation strategy is explained through the example of \(14 \times 12\) shown below:
Example \(\PageIndex{1}\):
Evaluate \(14 \times 12\).
Solution
1. Start with \(1 \) and the other number, and double them.
2. Keep doubling until the left-hand column is going to be bigger than the other number \(14\)
3. Since \(16 \) is bigger than \( 14\), stop there. See the table below.
| \(1\) | \(12\) |
| \(2\) | \(24\) |
| \(4\) | \(48\) |
| \(8\) | \(96\) |
| \(16\) |
4. Use the left column numbers to add to \(14 =8+4+2\) and add the corresponding numbers on the right \(96+48+24=168\) to get the product
● Why does this work? What you’re really doing is adding the appropriate doubles: \(96 \) is eight \(12\)s, \(48\) is four \(12\)s, and \(24\) is two \(12\)s. So when you add them you get \(14 \,12\)s.
The division is similar - demonstrates Egyptian knowledge that division and multiplication are reciprocal operations, like addition and subtraction.
Example \(\PageIndex{2}\):
Evaluate \(\dfrac{153}{21}\).
Solution
List the powers of \(2\) in a column. List the divisor beside the 1, and double it down the second column until the number approaches, but does not exceed the dividend.
| \(1\) | \(21\) |
| \(2\) | \(42\) |
| \(4\) | \(84\) |
- Subtract the largest possible right-column number from the dividend, noting which ones are used, until it is no longer possible:\(153 - 84 = 69, 69 - 42 = 27, 27 - 21 = 6\).
- The quotient is the sum of the left-hand column numbers associated with the subtracted quantities: So, since we used \(84, 42,\) and \(21,\) the quotient is \(4 + 2 + 1 = 7.\)
- The remainder is the result of the subtractions: \(7 \) remainder \(6\).
See Egyptian fraction.