1. Chapter 4 Numeration and Mathematical Systems
© 2008 Pearson Addison-Wesley.
All rights reserved

2. 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

3. Section 4-2 Chapter 1 Arithmetic in the Hindu-Arabic System
© 2008 Pearson Addison-Wesley. All rights reserved

4. Arithmetic in the Hindu-Arabic System
Expanded Form
Historical Calculation Devices
© 2008 Pearson Addison-Wesley. All rights reserved

5. © 2008 Pearson Addison-Wesley. All rights reserved
Expanded Form
By 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 reserved

6. Example: Expanded Form
Write the number 23,671 in expanded form.
Solution
© 2008 Pearson Addison-Wesley. All rights reserved

7. Distributive Property
For all real numbers a, b, and c,
For example,
© 2008 Pearson Addison-Wesley. All rights reserved

8. Example: Expanded Form
Use expanded notation to add 34 and 45.
Solution
© 2008 Pearson Addison-Wesley. All rights reserved

9. © 2008 Pearson Addison-Wesley. All rights reserved
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

10. 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

11. © 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

12. © 2008 Pearson Addison-Wesley. All rights reserved
Example: Abacus
Which number is shown below?
Solution
( ) (5 + 1) = 1706
© 2008 Pearson Addison-Wesley. All rights reserved

13. © 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

14. Example: Lattice Method
Find the product by the lattice method.
Solution
Set up the grid to the right. 3 8
© 2008 Pearson Addison-Wesley. All rights reserved

15. Example: Lattice Method
Fill in products 2 1 7 5 6 3 3 8
© 2008 Pearson Addison-Wesley. All rights reserved

16. Example: Lattice Method
Add diagonally right to left and carry as necessary to the next diagonal. 1 2 2 1 7 5 6 3 3
© 2008 Pearson Addison-Wesley. All rights reserved

17. Example: Lattice Method1 2 2 1 7 5 6 3 3
Answer: 30,172
© 2008 Pearson Addison-Wesley. All rights reserved

18. 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

19. © 2008 Pearson Addison-Wesley. All rights reserved
Napier’s Rods
Insert figure 2 on page 174
© 2008 Pearson Addison-Wesley. All rights reserved

20. Russian Peasant Method
Method of multiplication which works by expanding one of the numbers to be multiplied in base two.
© 2008 Pearson Addison-Wesley. All rights reserved

21. 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

22. Example: Nines Complement Method
Use the nines complement method to subtract 2803 – 647.
Solution
Step Step 2 Step 3 Step 4
© 2008 Pearson Addison-Wesley. All rights reserved