1. SECTION 4-3 Arithmetic in the Hindu-Arabic System Slide 4-3-1

2. ARITHMETIC IN THE HINDU-ARABIC SYSTEM Expanded Form Historical Calculation Devices Slide 4-3-2

3. EXPANDED FORM Slide 4-3-3 By using exponents, numbers can be written in expanded form in which the value of the digit in each position is made clear.

4. EXAMPLE: EXPANDED FORM Slide 4-3-4 Write the number 23,671 in expanded form. Solution

5. DISTRIBUTIVE PROPERTY Slide 4-3-5 For all real numbers a, b, and c, For example,

6. EXAMPLE: EXPANDED FORM Slide 4-3-6 Use expanded notation to add 34 and 45. Solution

7. DECIMAL SYSTEM Slide 4-3-7 Because our numeration system is based on powers of ten, it is called the decimal system, from the Latin word decem, meaning ten.

8. HISTORICAL CALCULATION DEVICES Slide 4-3-8 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.

9. ABACUS Slide 4-3-9 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.

10. EXAMPLE: ABACUS Slide 4-3-10 Which number is shown below? Solution 1000 + (500 + 200) + 0 + (5 + 1) = 1706

11. LATTICE METHOD Slide 4-3-11 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.

12. EXAMPLE: LATTICE METHOD Slide 4-3-12 Find the product by the lattice method. 7 9 4 3838 Solution Set up the grid to the right.

13. EXAMPLE: LATTICE METHOD Slide 4-3-13 Fill in products 2 1 2 7 1 2 5 6 7 2 3 2 7 9 4 3838

14. EXAMPLE: LATTICE METHOD Slide 4-3-14 Add diagonally right to left and carry as necessary to the next diagonal. 2 1 2 7 1 2 5 6 7 2 3 2 1 7 2 0 2 1 3

15. EXAMPLE: LATTICE METHOD Slide 4-3-15 Answer: 30,172 2 1 2 7 1 2 5 6 7 2 3 2 1 7 2 0 2 1 3

16. EXAMPLE: NINES COMPLEMENT METHOD Slide 4-3-16 Use the nines complement method to subtract 2803 – 647. Solution Step 1 Step 2 Step 3 Step 4