SECTION 4-3

• Arithmetic in the Hindu-Arabic System

Slide 4-3-1

ARITHMETIC IN THE HINDU-ARABIC SYSTEM

• Expanded Form• Historical Calculation Devices

Slide 4-3-2

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.

EXAMPLE: EXPANDED FORM

Slide 4-3-4

Write the number 23,671 in expanded form.

Solution 4 3 2 1 02 10 3 10 6 10 7 10 1 10

DISTRIBUTIVE PROPERTY

Slide 4-3-5

For all real numbers a, b, and c,

For example,

.b a c a b c a

4 4 4

4

3 10 2 10 3 2 10

5 10 .

EXAMPLE: EXPANDED FORM

Slide 4-3-6

Use expanded notation to add 34 and 45.

1 0

1 0

1 0

34 3 10 4 10

45 4 10 5 10

7 10 9 10 79

Solution

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.

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.

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.

EXAMPLE: ABACUS

Slide 4-3-10

Which number is shown below?

Solution1000 + (500 + 200) + 0 + (5 + 1) = 1706

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.

EXAMPLE: LATTICE METHOD

Slide 4-3-12

Find the product by the lattice method.38 794

7 9 4

3

8

SolutionSet up the grid to the right.

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

3

8

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 20

21

3

EXAMPLE: LATTICE METHOD

Slide 4-3-15

Answer: 30,172

2 1

2 7

1 2

5 6

7 2

3 2

1 7 20

21

3

EXAMPLE: NINES COMPLEMENT METHOD

Slide 4-3-16

Use 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 4