Slide 1Chapter 4: Numeration and Mathematical SystemsSlide 3Arithmetic in the Hindu-Arabic SystemExpanded FormExample: Expanded FormDistributive PropertySlide 8Decimal SystemHistorical Calculation DevicesAbacusExample: AbacusLattice MethodExample: Lattice MethodSlide 15Slide 16Slide 17Napier’s Rods (Napier’s Bones)Napier’s RodsRussian Peasant MethodNines Complement MethodExample: Nines Complement MethodChapter 4Numeration and Mathematical Systems © 2008 Pearson Addison-Wesley.All rights reserved© 2008 Pearson Addison-Wesley. All rights reserved4-2-2Chapter 4: Numeration and Mathematical Systems4.1 Historical Numeration Systems 4.2 Arithmetic in the Hindu-Arabic System 4.3 Conversion Between Number Bases 4.4 Clock Arithmetic and Modular Systems4.5 Properties of Mathematical Systems4.6 Groups© 2008 Pearson Addison-Wesley. All rights reserved4-2-3Chapter 1Section 4-2Arithmetic in the Hindu-Arabic System© 2008 Pearson Addison-Wesley. All rights reserved4-2-4Arithmetic in the Hindu-Arabic System•Expanded Form•Historical Calculation Devices© 2008 Pearson Addison-Wesley. All rights reserved4-2-5Expanded FormBy using exponents, numbers can be written in expanded form in which the value of the digit in each position is made clear.© 2008 Pearson Addison-Wesley. All rights reserved4-2-6Example: Expanded FormWrite the number 23,671 in expanded form.Solution4 3 2 1 02 10 3 10 6 10 7 10 1 10� + � + � + � + �© 2008 Pearson Addison-Wesley. All rights reserved4-2-7Distributive PropertyFor all real numbers a, b, and c,For example,( ) ( ) ( ).b a c a b c a� + � = + �( ) ( )( )4 4 443 10 2 10 3 2 10 5 10 .� + � = + �= �© 2008 Pearson Addison-Wesley. All rights reserved4-2-8Example: Expanded FormUse expanded notation to add 34 and 45. ( ) ( )( ) ( )( ) ( )1 01 01 0 34 3 10 4 10 45 4 10 5 10 7 10 9 10 79= � + �+ = � + �� + � =Solution© 2008 Pearson Addison-Wesley. All rights reserved4-2-9Decimal SystemBecause our numeration system is based on powers of ten, it is called the decimal system, from the Latin word decem, meaning ten.© 2008 Pearson Addison-Wesley. All rights reserved4-2-10Historical Calculation DevicesOne of the oldest devices used in calculations is the abacus. It has a series of rods with sliding beads and a dividing bar. The abacus is pictured on the next slide.© 2008 Pearson Addison-Wesley. All rights reserved4-2-11AbacusReading from right to left, the rods have values of 1, 10, 100, 1000, and so on. The bead above the bar has five times the value of those below. Beads moved towards the bar are in “active” position.© 2008 Pearson Addison-Wesley. All rights reserved4-2-12Example: AbacusWhich number is shown below?Solution1000 + (500 + 200) + 0 + (5 + 1) = 1706© 2008 Pearson Addison-Wesley. All rights reserved4-2-13Lattice MethodThe Lattice Method was an early form of a paper-and-pencil method of calculation. This method arranged products of single digits into a diagonalized lattice.The method is shown in the next example.© 2008 Pearson Addison-Wesley. All rights reserved4-2-14Example: Lattice MethodFind the product by the lattice method.38 794�7 9 438SolutionSet up the grid to the right.© 2008 Pearson Addison-Wesley. All rights reserved4-2-15Example: Lattice MethodFill in products2 12 71 25 67 23 27 9 438© 2008 Pearson Addison-Wesley. All rights reserved4-2-16Example: Lattice MethodAdd diagonally right to left and carry as necessary to the next diagonal.2 12 71 25 67 23 21 7 20213© 2008 Pearson Addison-Wesley. All rights reserved4-2-17Example: Lattice MethodAnswer: 30,1722 12 71 25 67 23 21 7 20213© 2008 Pearson Addison-Wesley. All rights reserved4-2-18Napier’s Rods (Napier’s Bones)John Napier’s invention, based on the lattice method of multiplication, is often acknowledged as an early forerunner to modern computers. The rods are pictured on the next slide.© 2008 Pearson Addison-Wesley. All rights reserved4-2-19Napier’s RodsInsert figure 2 on page 174© 2008 Pearson Addison-Wesley. All rights reserved4-2-20Russian Peasant MethodMethod of multiplication which works by expanding one of the numbers to be multiplied in base two.© 2008 Pearson Addison-Wesley. All rights reserved4-2-21Nines Complement MethodStep 1 Align the digits as in the standard subtraction algorithm.Step 2 Add leading zeros, if necessary, in the subtrahend so that both numbers have the same number of digits.Step 3 Replace each digit in the subtrahend with its nines complement, and then add. Step 4 Delete the leading (1) and add 1 to the remaining part of the sum.© 2008 Pearson Addison-Wesley. All rights reserved4-2-22Example: Nines Complement MethodUse the nines complement method to subtract 2803 – 647.Solution 2803 2803 2803 2155647 0647 +9352 1 12,155 2156- - +Step 1 Step 2 Step 3 Step