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IDENTIFYING PLACE VALUES
Jump right to the exercises!
See the best ALGEBRA PINBALL time for this exercise
The concepts for this exercise are summarized below.
For a complete discussion, read the text.
The system that we use to represent numbers is wonderfully efficient and simple.
Large numbers can be represented with very few symbols in a neat and organized way.
The method rests on two concepts, base and place value,
which are briefly discussed in the next few paragraphs.
The concepts are introduced using base three for simplicity.
Then, the concepts are extended to our base ten number system.
The concept of base is best represented by grouping.
Imagine a factory where objects are being packaged for shipment around the world.
Workers are instructed to "bundle things up" every time they get a group of three.
Here are the names for the packaging that they use:
- a set of three objects is called a packet
- a set of three packets (which is 3 · 3 = 9 objects) is called a box
- a set of three boxes (which is 9 · 3 = 27 objects) is called a carton
Notice that a packet holds 31 = 3 objects;
a box holds 32 = 9 objects; and
a carton holds 33 = 27 objects.
Suppose a shipment of 46 objects is to go out to Lenox, Massachusetts.
How would these 46 objects get bundled?
First, bundle up one carton, leaving 46 - 27 = 19 objects remaining.
Then, bundle up two boxes, leaving 19 - 2·9 = 1 object remaining.
Thus, 46 items get bundled into one carton, two boxes, no packets, and one object, as shown below:
The workers use a shorthand to keep track of how orders get bundled. The record for
this shipment of 46 looks like this:
The workers read this aloud as "one, two, zero, one; base three".
Each position (place value) represents a power of 3 :
the right-most place value is 30 = 1 (objects);
one to the left is 31 = 3 (packets);
one more to the left is 32 = 9 (boxes); and
one more to the left is 33 = 27 (cartons).
(Here, you're getting a preview of exponent notation, which will be explored in a future section.)
Of course, bigger shipments would require more place values, and more names for packaging types.
This is called a base three number system, because things are bundled in groups of three.
The number system that we use is called the base ten number system.
It uses identical concepts, except that things are bundled in groups of ten.
Here are the first few place values
and their names in our base ten number system:
Notice that the right-most place value is 1 , and you just keep multiplying by 10 as you
move to the left.
For example, the base ten number 23,487 represents:
23,487
= 2·104 + 3·103 + 4·102 + 8·10 + 7·1
= 2·10,000 + 3·1,000 + 4·100 + 8·10 + 7
= 2 ten-thousands + 3 thousands + 4 hundreds + 8 tens + 7 ones
This exercise gives you practice recognizing place values.