The Hindu-Arabic System

•The Hindu-Arabic
system is a base ten positional number system.

•By this we mean
that every numeral in a Hindu-Arabic number has a place value of some power of
ten.

–For example: In the number 1,432 the numeral 2 is in the
10^{0} position, the numeral 3 is in the 10^{1} position, the
numeral 4 is in the 10^{2} position and the numeral 1 is in the 10^{3}
position.

•Any number can be
written in __Expanded Form__ with the use of exponents as follows:

–The number 1,432
written in expanded form is:

•1 × 10^{3}
+ 4 × 10^{2} + 3 × 10^{1} + 2 × 10^{0} or equivalently

•1 × 1000 + 4 ×
100 + 3 × 10 + 2 × 1

•Example: write
the following numbers in expanded form:

–13,256

–2,398

–643

•Example: Use
expanded form to add or subtract as indicated:

–253 + 45

–452 − 123

Arithmetic Methods

•Man has come up
with many methods to make arithmetic simpler, several such methods are listed
below; we will discuss each one individually.

•Method of Nines
Complements

•Lattice Method

•Russian Peasant
Method

•Egyptian
Algorithm

The Method of Nines Complements

•The Nines
complements method is a method that subtracts numbers by adding the “nines
complement.”

•The Nines
complement of a number (0-9) is 9 minus the number.

–Examples: The
nines complement of 5 is 4.

The
nines complement of 3 is 6.

The
nines complement of 0 is 9.

•To write the nines
complement of a number, simply write the nines complement of each digit.

–Example: The nines complement of 23,487 is
76,512

Using Nines Complement Method

•To use the nines
complement method to subtract two numbers:

–Replace the second number with its nines
complement.

–Add the two numbers.

–Delete the first digit from the sum and add
its value to the remaining part of the sum.

•This is the
answer to the original subtraction problem.

•Example: Subtract using nines complement method: 23,176
– 15,897

564,843
– 98,764

Lattice Method for Multiplication

•The lattice
method arranges products of single digits into a diagonal lattice as the next
example will show.

•Multiply using
the lattice method:

–23 x 45

–348 x 67

–4369 x 342

Russian Peasant Method

•The Russian
Peasant Method is an algorithm for multiplication.

–To multiply numbers using the Russian Peasant
Method do the following:

•Write the two
numbers to be multiplied at the head of two columns.

•Divide repeatedly
the numbers in the first column by 2, ignoring remainders, until one is
obtained.

•Multiply
repeatedly the numbers in the second column by 2 as many times as you divided
the first column, so that the two columns have the same amount of numbers in
them.

•Add all the
numbers in the second column that correspond to odd numbers in the first
column.

•The sum obtained
in step 4 is the product of the original two numbers.

Examples

•Multiply using
the Russian Peasant Method:

–32 x 15

–25 x 12

–125 x 13

Egyptian Algorithm

•To use the
Egyptian Algorithm for multiplication do the following:

–Make two columns of numbers; start with the
number one in the left most column and keep doubling
it. In the right column start with one
of the numbers to be multiplied and double it.

–When the there are enough numbers in the first
column so that the sum of the factor not written in the second column is
obtained, stop doubling.

–Find the numbers in the first column that add
up to the factor not listed in the second column, and add the corresponding
numbers in the second column.

–This sum is the product of the original two
numbers.

Examples

•Multiply using
the Egyptian Algorithm:

–13 x 25

–21 x 65

–54 x 23