Chapter 4 Numeration and Mathematical Systems

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Chapter 4 Numeration and Mathematical Systems

Chapter 4: Numeration and Mathematical Systems
4.1 Historical Numeration Systems 4.2 Arithmetic in the Hindu-Arabic System 4.3 Conversion Between Number Bases 4.4 Clock Arithmetic and Modular Systems 4.5 Properties of Mathematical Systems 4.6 Groups © 2008 Pearson Addison-Wesley. All rights reserved

Section 4-2 Chapter 1 Arithmetic in the Hindu-Arabic System

Arithmetic in the Hindu-Arabic System

Example: Expanded Form

Distributive Property

Example: Expanded Form

Decimal System Because 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 reserved

Historical Calculation Devices
One 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 reserved

Abacus Reading 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 reserved

Lattice Method The 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 reserved

Example: Lattice Method

Example: Lattice Method

Example: Lattice Method

Example: Lattice Method

Napier’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 reserved

Russian Peasant Method

Nines Complement Method
Step 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 reserved

Example: Nines Complement Method