Section 2 3 The HinduArabic System HinduArabic System

Section 2. 3 The Hindu-Arabic System

Hindu-Arabic System Set of symbols 1 2 3 4 5 6 7 8 9 0 place holder Characteristics • base ten • additive • has place value • has a unique symbol for each number less than the base. (on the next slide you will see that other systems just repeat the “one” symbol to represent the numbers “two”, “three”, or even beyond)

Different symbols for the small numbers one two three I II III • • • • 1 2 3 Tally Egyptian Roman Babylonian Mayan Chinese Hindu-Arabic As you can see, only the Hindu-Arabic system uses a different symbol for “three”, all other systems use the symbol for “one” three times, (which is very intuitive, but not compatible with place value structure) and hence the Hindu-Arabic system is difficult for children to learn.

Section 2. 3 Hindu-Arabic System has an important feature called the place value Example: hundreds thousands tens ones 2023 The same symbol means two different quantities when placed in two different positions.

Hindu-Arabic System has an important feature called the place value Example: hundreds thousands tens ones 5614 For this particular symbol, the • face value is six • place value is one hundred • actual value is six hundred

Hindu-Arabic System has an important feature called the place value Example: hundreds thousands tens ones 5614 The Expanded form of 5614 is 5× 1000 + 6× 100 + 1× 10 + 4× 1

Naming Hindu-Arabic Numerals 123, 456, 789, 987, 654, 321 quadrillion trillion billion million thousand

Base Ten Blocks Block =1000 a Small cube = 1 a Flat =100 a Long =10

5614 means 5 Thousands 6 Hundreds 1 Ten 4 Ones and is represented by

Hindu-Arabic System Advantages • a relatively small finite set of symbols is used • very concise (a few symbols rep. a big number) • the length of an expression is proportional to the size of the number being represented. This facilitates calculations. Disadvantages • the place value structure is sophisticated and difficult to master. • students have to learn many symbols to read or write even relatively small numbers (eg. compare this with the Roman system)

To understand potential difficulties in learning Hindu-Arabic system and to assist the development of better teaching methods, we introduce the Base Five Hindu-Arabic system Symbols required: 1, 2, 3, 4, 0

Base Five Hindu-Arabic system Symbols required: 1, 2, 3, 4, 0 A list of small numbers one two three four five six seven eight 1 five 2 five 3 five 4 five 10 five 11 five 12 five 13 five nine 14 five ten 20 five eleven 21 five twelve 22 five. . . twenty four 44 five twenty five 100 five

Base Five Hindu-Arabic System Place values Twenty five One hundred twenty five 2023 The expanded form of 2023 Five is 2× 53 + 0× 52 +2× 5 + 3× 1 five one

Physical representations of numbers in base Five Sticks and bundles a stick = one a bundle = five a super-superbundle = five superbundles = twenty five bundles = one hundred twenty five a superbundle = five bundles = twenty five

Examples: What is the base five numeral for the following number of sticks? Ans: 13 five How about the following? Ans: 102 five Why do we need the 0?

Examples: What is the physical representation for 132 five ? How many sticks are there? Ans: twenty five + three×five + two = forty two. Convert thirty two to a base five numeral. Ans: 112 five. Convert eighty three to a base five numeral Ans: 313 five.

Base Five Blocks a (small) cube = one a long = 5 a flat = 25 a block = 53 = 125

Base Five Blocks Examples: 23 five = = five + three = thirteen 1204 five = = hundred twenty five +twenty five + four = one hundred seventy nine.

Trades A trade is an exchange of objects of equal value. Example: • Trade five small cubes for one long • Trade five longs for one flat

Trades Or • Trade five flats for one block

Base Five Numerals We need to perform trades before we can write down the base five numerals for these quantities. Base Three Numerals We need to perform trades before we can write down the base three numerals for these quantities.

Shepherd boy and his Sheep In the daytime

Shepherd boy and his Sheep In the night-time