Download duplation and mediation

Section 4.5
Early
Computation
Methods
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
Other computation methods:
Duplation and mediation
Lattice method
Napier’s rods
4.5-2
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Early Civilizations
Early civilizations used a variety of
methods for multiplication and division.
Multiplication was performed by
duplation and mediation, by the lattice
method, and by Napier’s rods.
4.5-3
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Duplation and Mediation
Know as Russian Peasant Multiplication
Duplation refers to doubling a number
Mediation refers to halving a number
Similar to ancient Egyptian method
described in Rhind Papyrus
4.5-4
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 1: Using Duplation and
Mediation
Multiply 19 × 17 using duplation and
mediation.
Solution
Write 19 and 17 with a dash between
19 – 17
Divide the number on the left, 19, by 2
Drop the remainder
Place the quotient, 9, under the 19
4.5-5
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 1: Using Duplation and
Mediation
Solution
19 – 17
9
Double the number on the right, 17, to
obtain 34 place it under the 17
19 – 17
9 – 34
Continue this process until 1 appears in
the left-hand column
4.5-6
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 1: Using Duplation and
Mediation
Solution
19 – 17
9 – 34
4 – 68
2 – 136
1 – 272
Cross out all the even numbers in the
left-hand column and the corresponding
numbers in the right-hand column
4.5-7
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 1: Using Duplation and
Mediation
Solution
19 – 17
9 – 34
4 – 68
2 – 136
1 – 272
Add remaining numbers in right-hand
column: 17 + 34 + 272 = 323
19 × 17 = 323
4.5-8
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Try This
Multiply 27 x 13
4.5-9
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
The Lattice Method
The Lattice method is also referred to
as the gelosia method.
The name comes from the use of a
grid, or lattice, when multiplying two
numbers.
4.5-10
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
The Lattice Method
This method uses a rectangle split into
columns and rows with each newlyformed rectangle split in half by a
diagonal.
4.5-11
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 2: Using Lattice
Multiplication
Multiply 312 × 75 using lattice
multiplication.
Solution
Construct a rectangle consisting of 3
columns and 2 rows
Place the 312 above the boxes
Place 75 on the right of the boxes
Place a diagonal in each box
4.5-12
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 2: Using Lattice
Multiplication
Solution
3
1
2
7
5
4.5-13
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 2: Using Lattice
Multiplication
Solution
Complete each box by multiplying the
number on the top of the box by the
number on the right of the box
Place the units digit of the product
below the diagonal
Place the tens digit of the product
above the diagonal
4.5-14
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 2: Using Lattice
Multiplication
Solution
3
2
1
4.5-15
1
1
5
0
0
2
7
5
1
1
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
4
0
7
5
Example 2: Using Lattice
Multiplication
Solution
Add diagonals as shown
carry tens digits to next diagonal
3
1
2
2 2
1
3 1
5
4
4.5-16
0
7
0
5
0
1
4
1
0
0
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
7
5
Example 2: Using Lattice
Multiplication
Solution
Read the answer down the left-hand
column and along the bottom
3
1
2
0
1
2 2
7
1
7
4
3 1
5
4
4.5-17
0
5
0
1
0
5
0
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Therefore,
312 × 75
= 23, 400
Try This
Multiply 237 x 73
4.5-18
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Homework
4.5-19
Copyright 2013, 2010, 2007, Pearson, Education, Inc.