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 10⁰ position, the numeral 3 is in the 10¹ position, the numeral 4 is in the 10² position and the numeral 1 is in the 10³ 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 × 10³ + 4 × 10² + 3 × 10¹ + 2 × 10⁰ 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