Hindu-Arabic and Early Positional Systems 4.1 Our Number System Hindu-Arabic and Early Positional Systems

The Hindu-Arabic number system uses 10 symbols for digits that were developed in India and then used by the Arabs. The symbols were popularized by Leonardo de Pisa in a book called Liber Abaci in 1202. Prior to 1202 the most popular system was the Roman system using M, C, X, etc. for digits. The advantage of the Hindu-Arabic system is the positional notation, digits have different values depending on their position. 123,456 325,417 The 5 here means five tens. The 5 here means five thousands.

723,456 (digit times place value plus digit times place The 4 here means four hundreds. The 2 here means two ten thousands. The 6 here means six ones. 723,456 The 7 here means seven hundred thousands. The 5 here means five tens. The 3 here means three thousands. When we write a number in expanded notation, the place values are made explicit. (digit times place value plus digit times place value plus digit times place value …)

Write the number 4,873 in expanded notation. Usually we write the place values using powers of 10 instead of writing them out. The exponent is the same as the number of zeroes following the one.

287,090 Write the number in standard Hindu-Arabic form. Since we use ten digits and powers of ten for the place value, we call this base 10. There are other possibilities. Five has been used as a base. Twenty has also been popular. Why have five ten and twenty been the most common bases? One group of people use base two. Why?

The Babylonians used a place value system of numbers as early as 2000 B.C.. They used a system with two symbols to produce their “digits” and used a place value of sixty. The effect of the Babylonian system is still with us. We use base 60 for time. There are 60 seconds in a minute and 60 seconds in an hour. The Babylonian system is also responsible for our measurement of angles. There are 360 degrees in a circle. The two symbols were V for one and for ten. V VVV V V V V V would be 35 VVVVV V V V V V V V V V would be 59, this is the largest “digit” in base 60. 62 would be written as V V V. There is one 60 and two ones. In expanded form:

The two symbols were V for one and for ten. VVV V V V V V would be 35 VVVVV V V V V V V V V V would be 59, this is the largest “digit” in base 60. 62 would be written as V V V. There is one 60 and two ones. In expanded form:

A Babylonian System The Babylonians used the symbol for “one” and the symbol < for “ten.” They used a positional system with powers of 60. Example: \/ \/ < \/ < < \/ \/ (1+1) x 602 + (10+1) x 601 + (10+10+1+1) x 1 = (2 x 3600) + (11 x 60) + (22 x 1) = 7200 + 660 + 22 = 7882

The place value gets large rapidly and the “digits” become long. To write a Babylonian number in expanded form, use the digit times the place value technique. VVV V V VVVV V V V V This is 32 x 3,600 This is 12 x 1 This is 41 x 60

Write the Babylonian number in Hindu-Arabic form. VVV V V V V V VVVVV V VV V V V V This is 35 x 3,600 This is 24 x 1 This is 51 x 60

Write the Babylonian number in Hindu-Arabic form. VV V V V VVV V V This is 23 x 60 This is 32 x 1 The Babylonians did not have a symbol for zero. This lack causes a problem.

VV V V V VVV V V This could be 23 x 60 It could also be 23 x 3,600 or 23 x 216,000 This could be 32 x 1 It could also be 32 x 60 or 32 x 3,600 Without a place holder such as zero, we can not be sure. It might be 1,412. Possibly the value is 82,832 or 4,968,032 or 4,969,920 or 5,083,200. The context would help, but there are always possible ambiguities.

First, what is the highest power of 60 needed to write the number? To write 10,849 as a Babylonian number, we will need to divide and conquer. First, what is the highest power of 60 needed to write the number? 216,000 is too large. 3600 will be the highest power of 60 that we need. It will be a three digit Babylonian number. Divide 10,849 by 3600 to find the digit in the 3600 place. What is the highest power of 60 needed to write the remainder? 60 is too large. One will be the highest power of 60 that we need. There will be nothing in the sixty’s place. The number can be written as: VVVV V V V V V V V V V V V V

Write 20,063 as a Babylonian number. First, what is the highest power of 60 needed to write the number? 216,000 is too large. 3600 will be the highest power of 60 that we need. It will be a three digit Babylonian number. Divide 20,063 by 3600 to find the digit in the 3600 place. What is the highest power of 60 needed to write the remainder? 60 will be the highest power of 60 that we need. Divide 2,063 by 60 to find the digit in the 60 place. The remainder will go in the ones place. VVV V V V V V V V V V VV V V V

The Mayans were the first to use a symbol for zero The Mayans were the first to use a symbol for zero. There system is a modified base 20. 1 2 3 4 5 6 7 8 9 11 12 13 14 10 15 16 17 18 19

The place values for the Mayan system are: 6 x 360 0 x 20 12 x 1

Write the Mayan numeral in Hindu-Arabic. 2 x 7,200 16 x 360 8 x 20 10 x 1

Write the Mayan numeral in Hindu-Arabic. 5 x 144, 000 0 x 7,200 0 x 360 11 x 20 13 x 1

Write the Hindu-Arabic numeral 19,452 in Mayan. Remember Write the Hindu-Arabic numeral 19,452 in Mayan. Remember. Divide and conquer. First, what is the highest place value needed to write the number? 144,000 is too large. 7,200 will be the highest place value that we need. It will be a four digit Mayan number. Divide 19,452 by 7,200 to find the digit in the 7,200 place. What is the highest place value needed to write the remainder? 360 will be the highest place value that we need. Divide 5,052 by 360 to find the digit in the 360 place. Zero will go in the twenty’s place. The remainder will go in the ones place.

Let’s not! Write the Hindu-Arabic numeral 6,575 in Mayan. First, what is the highest place value needed to write the number? 7,200 is too large. 360 will be the highest place value that we need. It will be a three digit Mayan number. Divide 6,575 by 360 to find the digit in the 360 place. What is the highest place value needed to write the remainder? 20 will be the highest place value that we need. Divide 90 by 20 to find the digit in the 20 place. The remainder will go in the ones place. Let’s convert the Babylonian numeral V V V V to Mayan! VV Let’s not!