1.1+Place+Value

Section 1.1 - Place Value
The video below shows a coin sorter - a cleverly built one that is open and you can see exactly how it works. Before watching the video, predict what the result will look like after a handful of coins are dropped into the top end. See if the result more or less matches up to what you expect.

media type="youtube" key="vZaRgMOo-Ig?version=3" height="390" width="640"

Shifting gears, just for a minute, consider that we have only ten digits (0-9) in our numbering system, yet countless numbers can be written just by the intentional arrangement of those digits. Include a negative sign and perhaps a well-placed decimal point, and virtually any number could be written.

Consider the ages of these two people. The same two digits are used in each person's age, but they are not the same number, and do not mean the same thing. Putting the "1" before the "7" rather than after it causes it to take on a different -- in fact ten times its value otherwise.



In our "base ten" numbering system, each column, or "place" to the left that a digit is placed multiplies it by ten. The name of each place also refers to this effect of multiplying by tens. One thousand is ten times more than one hundred, and so forth.


 * The Importance of Zero**

Zero has no value by itself, but is very important in the place value system to allow other digits to line up in the correct places. The zero is not mentioned when the number is said. So, for example, 207 is read as two hundred seven.


 * Periods**

Starting at the right side (where the decimal point would be located) and moving to the left, each group of three places is known as a period. These periods are separated by commas in America. (The name might explain why periods are used to separate them in other parts of the world.) Within each period the number is essentially seen as a one-, two-, or three-digit number, followed by the period name. For example, 245 is two hundred forty-five. If it were in a different period, such as 245,000, it would be two hundred forty-five thousand.




 * Writing Place Value Numbers**

When a number is written, which is becoming less common for larger numbers, the periods names are important and the commas between them are helpful.

There are ** four rules ** to remember when writing a number out in word form:
 * 1) Include the commas where the period breaks occur.
 * 2) Include the name of the period before each comma (except the "Ones" period).
 * 3) Do not write "and" unless it is referring to the decimal point.
 * 4) Hyphenate all numbers from 21 to 99.

So, the number 3,502,029,343 becomes: three billion, five hundred two million, twenty-nine thousand, three hundred forty-three.


 * Expanded Form for Numbers**

Normally we use the "place value form" for numbers, which is just the digits written out, separated by commas into their periods. However, there is another form, called "Expanded Form", which uses each place more explicitly. Under the expanded form, the number 34,803 would be written as:

30,000 + 4,000 + 800 + 3

Notice that a place that does not have a value (uses a zero as a placeholder) is not included. The expanded form is not used often.


 * Whole Numbers**

It is easier to define what "whole numbers" are by explaining what they aren't. If you have only a part of something it isn't whole, so half of a peach is not a whole peach. In numbers, a part is generally represented as a fraction or as a decimal number. So fractions and decimal numbers are not considered whole numbers. Zero is included, but negative numbers are not, though that is less intuitive.

The set of whole numbers is 0, 1, 2, 3, 4, and so on -- forever. Elipses, or three dots, is used to represent that the counting goes on in the same pattern forever in a particular direction. You will usually see it written in set notation as:

W = {0, 1, 2, 3. . . }

There is another set of numbers called "Natural Numbers" or "Counting Numbers", which is the same, except that zero is not included. N = {1, 2, 3, 4. . . }

**Where to from here?**
1.1 Place Value 1.2 Rounding and Estimating Numbers 1.3 Addition 1.4 Subtraction 1.5 Multiplication 1.6 Division 1.7 Exponents 1.8 Order of Operations 1.9 Prime Numbers
 * Chapter One Review**
 * Chapter One Homework**