Whole numbers and numeration

Math 123

Counting

•Why do we count the way we do? •Have humans always been counting this

way? •Do all humans count in base 10? •Who were the first people to use zero?

Early counting history• In Egypt, from about 3000 BCE, records survive in

which 1 is represented by a vertical line and 10 is shown as ^.

• The Babylonians, around 1750 BCE, use a numerical system with 60 as its base. Their base of 60 survives even today in the 60 seconds and minutes of angular measurement, in the 180 degrees of a triangle and and in the 360 degrees of a circle. The Babylonians introduce the place-value concept.

• Another civilization, that of the Maya, independently arrives at a place-value system - in their case with a base of 20. They are thought to be the first to have a symbol for zero as it is used today, before 36 BCE. Like Babylonians, they do not have separate digits up to their base figure.

Ancient numeration systems in actionHere is an applet that converts between different

numeration systems.

Hindu-Arabic numerals• The Indians were the first to use a symbol for each digit.

They used a dot or small circle when the place in a number has no value, and they gave this dot the name sunya, meaning 'empty'. Zero makes its appearance around the 3rd century BCE, while the entire system was fully evolved by about 800 CE.

About two centuries later the Indian digits reached Europe in Arabic manuscripts, becoming known as Arabic numerals. Several more centuries passed before the ten Arabic numerals replaced the system inherited in Europe from the Roman empire.

(http://www.historyworld.net/wrldhis/PlainTextHistories.asp?historyid=ab34)

Why Arabic numerals?• Muḥammad ibn Mūsā

al-Khwārizmī (lived sometime between 800 and 850)

• “House of Wisdom” in Baghdad, the center of scholarship

• De numero indorum (Concerning the Hindu Art of Reckoning) – the new notation came to be known as that of al-Khwarizmi, or algorismi

Place value

Think about the following questions: •What is place value?•Which properties does a place value

numeration system have?•What are the advantages of this type of

system? •What is the base of a system?•Why do we use a base 10 system?

Virtual manipulatives

•Google “National library of virtual manipulatives” or enter nlvm.usu.edu

•Allow Java to be used, and say no to updates.

•Go to the “Number and operations” category and choose “Base blocks.”

•Change your base to any number other than 10.

Virtual manipulatives You will choose different bases to work with, but in any case, the names for the blocks are:

• Unit• Long• Flat• Block Learn how to count in these bases. Become

acquainted with the blocks. They are crucial for understanding place value systems, as well as operations with whole numbers.

Base 5

•Now let’s just focus on base 5. ▫What comes after 245, 4445, 12345?

▫What comes before 405, 3005, 123405?▫Is there 50 in base 5?

What is a base?

•What is going on when we go from 245 to 305, both in terms of blocks and in terms of numbers? How is this similar to going from 29 to 30 in base 10?

•What is a long called in every base?•No matter which base you are in, you will

say that you are in base 10. Why?

Place value

Having worked in bases 2, 3, 4, 5, 6, 7, and 10, which all have place value, think about the following questions:

•Which properties does a place value numeration system have?

•What are the advantages of this type of system?

•What is the base of a system?•Why do we use a base 10 system?

Properties of place value systems•No tallies. Any amount can be expressed

using a finite number of digits (ten in the case of our system).

•The value of each successive place to the left is (base)*the value of the previous place. In our system the base is 10. The values of the places are:… 100,000 10,000 1000 100 10 1

•Expanded form: every number can be decomposed into the sum of values from each place. In the case of our system: 234 = 2*100 + 3 *10 + 4*1.

•The concept of zero.

Why base 10?

•Because we have ten fingers. It is actually not the most convenient base for computation. Base 8 or 16 would be more convenient.

What is the base?

•The easiest way to think about it: the number of units in a long. It is the number of units you trade in for the next place value, the long.

Why study different bases?

•Because you have been using the base 10 system for 15+ years. When you use the base 5 system, your experience is similar to the experience of a five-year old. Furthermore, properties of place value systems can be better seen in an unfamiliar system.

•Base 2 and base 16 are commonly used in computer science.

Difficulties with place value

•Examples:▫Twenty-nine, twenty-ten, twenty-eleven▫Twenty-nine, thirty-one

•Children do not necessarily understand the concept of tens and ones; for example, it may not be clear to them that eleven is ten plus one;

•Difficulties with operations (will will see many examples of this).