2.2 Sets, Whole Numbers, and Numeration: Whole numbers and Numeration

Objective: Students distinguish between the uses of numbers.

But what precisely do we mean by the whole numbers?

A number is an idea, or an abstraction, that represents a quantity. The symbols that we see, write, or touch when representing numbers are called numerals. There are three common uses of numbers.

• The most common use of whole numbers is to describe how many elements are in a finite set. When used in this manner, the number is referred to as a cardinal number.

• A second use is concerned with order. For example, you may be second in line, or your team may be fourth in the standings. Numbers used in this way are called ordinal numbers.

• Finally, identification numbers are used to name such things as telephone numbers, bank account numbers, and social security numbers. In this case, the numbers are used in a numeral sense in that only the symbols, rather than their values, are important.

Example 1 Identify how the number is used in the sentence: cardinal, ordinal, or identification. a) Judy came in second in the race.

b) A dog is an animal with four legs.

c) I live in apartment five.

Ordering of whole numbers

If 𝑎 = 𝑛(𝐴) and 𝑏 = 𝑛(𝐵). Then 𝑎 < 𝑏 (read “a is less than b”) or 𝑏 > 𝑎 (read “b is greater than a” if A is equivalent to a proper subset of B.

Why is 3 smaller than 5?

1) By counting “one, two, three, four, five…”

2) 2 sets where one has 3 items and the other has 5 items.

3) A whole number line, numbers increase from left to right.

Example 2

Determine the greater of the two numbers 4 and 7 in three different ways.

Objective: Use other numeration systems

Number Systems The Tally Numeration System

The Babylonian Numeration System

The Egyptian Numeration System

Roman Numerals: The Roman numeration system is a subtractive system since it permits

simplifications using combinations of basic Roman numerals

Mayan Numerals: The Mayan numeration system, which developed between 300 and 900 c.e.,

was a vertical place-value system, and it introduced a symbol for zero.

Example 3

Place values in the Mayan Numeral System

Example 4 What number do each of the Mayan numerals below represent?

Example 5 Convert to Mayan

a) 312

b) 7908

Try on your own Express the following Mayan numerals in our numeration system

2.2 Homework: Section 2.2 Problem Set B page 66-67: #1-3, 5-6, 8, 10, 27

Analyze Student Thinking

2.27 Blanca says that since zero means nothing, she doesn't have to use a numeral for zero in the Mayan numeration system.

Instead, she can simply leave a blank space. Is she correct? Explain.