Session One

Welcome to the first section of MTH140, Number bases.  In this session you will learn about the place values in the decimal system as well as how to convert decimal numbers in standard notation to expanded notation.

Begin this session by watching the video tape.  If you have purchased a textbook, follow along in your book. If you are using the Internet version, follow along on the webpage. If you have any difficulties, contact your instructor.  After the explanation of place values, turn off the video player and try to work the practice problems.  Once you have worked the problems, check your answers in the answer key listed in the back of your book or click on the Answer Key link on the menu above.

Section One

 Place Values In The Decimal System

The system that is most familiar is the decimal system. This system is based on 10 digits. The digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are used in the decimal system. When counting, once all of the digits have been used, a combination of digits must then be used. As digits are added, place value increases. For example, the "number" after 9 is 10. Two digits are used in the number 10; the digit placement in the number tells us the "size" of the number. The digits are used sequentially when we build numbers. The number after 19 is 20. This is because the digit on the far right has reached the maximum of value of 9 and we must begin to "reuse" values in this position.

 1.1~ Converting from Standard Notation to Expanded Notation

Each value of a number is determined by the position of each digit in the number. The "place" of the digit has a name. In the number 12345, we refer to the position where the 5 is as the "units place." The 4 is in the "tens" place. The 3 is in the "hundreds" place. The 2 is in the "thousands place." The 1 is in the "ten-thousands place." The number 12345 is in standard notation. If we looked at the number 12345 and wrote it in a different way, we might write

 

Digit	x	Multiple of Ten	=	Value
5	x	1	=	5
4	x	10	=	40
3	x	100	=	300
2	x	1000	=	2000
1	x	10000	=	10000
Total Value			=	12345

 

Notice the #'s in the multiple of ten column are all powers of ten.

Recall that the number 1 could be written as ten to the zero power (10⁰); 10 could be written as ten to the first power (10¹); 100 could be written as ten to the second power (10²); 1000 could be written as ten to the third power (10³); and 10000 could be written as ten to the fourth power (10⁴).

If we wrote the number 12345 without using the chart, it would be written as:

5 x 1 + 4 x10 + 3 x 100 + 2 x 1000 + 1 x 10000

Another way of writing this would be:

5 x 10⁰ + 4 x 10¹ + 3 x 10² + 2 x 10³ + 1 x 10⁴

We call this expanded notation.

 

Figure 1 ~ 1.1

Place Values for Decimal Number 12345

 

1    2    3    4     5

­|   ­  |   ­   |   ­ |     |____________10⁰

|     |      |    |_____________ 10¹

|     |      |______________10²

|     |________________10³

|_________________10⁴

Example: Convert 472691 to expanded notation.

Solution: First, identify the value of each digit. The digit 1 is in the units place; the digit 9 is in the tens place; the digit 6 is in the hundreds place; the digit 2 is in the thousands place; the digit 7 is in the ten-thousands place; and the digit 4 is in the hundred-thousands place.

 

Table 1 ~ 1.2

Digit	x	Multiple of Ten (Power of Ten)	=	Value
1	x	1 (or 100)	=	1
9	x	10 (or 101)	=	90
6	x	100 (or 102)	=	600
2	x	1000 (or 103)	=	2000
7	x	10000 (or 104)	=	70000
4	x	100000 (or 105)	=	400000
Total Value			=	12345

 

If we wrote the number without using the chart, the number would be

1 x 10⁰ + 9 x 10¹ + 6 x 10² + 2 x 10³ + 7 x 10⁴ + 4 x 10⁵

The number is now in expanded notation.

 

Figure 1 ~ 1.2

Place Values for the decimal number 472691

4       7        2         6        9       1

­|      ­  |      ­  |      ­    |   ­      |       |­ ___________10⁰

|        |        |          |         | _____________10¹

|        |        |          |_________________10²

|        |        |____________________10³

|        |_______________________10⁴

|__________________________10⁵

 

1.1 ~ Practice Problems

Directions: Turn off your VCR.  Change the following numbers from standard notation to expanded notation.  When you have finished working the problems, check your answers in the back of the book or click here to view the Answer Key. Turn on the video to see the solutions to the practice problems worked out on the board by a teacher.

1. 286
2. 10568
3. 23
4. 7
5. 10586374

1.1 ~ Homework Problems

Directions: Turn off the video.  Convert the following numbers from standard notation to expanded notation.  Keep your work in a notebook to be turned into your instructor at the conclusion of the course.  When you have finished working the problems, check your answers in the back of the book or click here to view the Answer Key. If you have any questions, contact your instructor.

1. 156
2. 10289
3. 6078937
4. 3
5. 5169
6. 62
7. 484960
8. 10000
9. 3795667
10. 9371

Continuing Section One

Repeat the first steps on this page.

1.2- Converting from Expanded Notation to Standard Notation

We can change numbers that have been written in expanded notation to standard notation by reversing the process. Suppose we have the notation

5 x 10⁰ + 4 x 10¹ + 3 x 10² + 2 x 10³ + 1 x 10⁴

and we want to convert to standard notation. We might begin by writing the powers of ten in another form.

5 x 1 + 4 x 10 + 3 x 100 + 2 x 1000 + 1 x 10000

Table 1~ 2.1

Digit	x	Multiple of Ten	=	Value
5	x	1	=	5
4	x	10	=	40
3	x	100	=	300
2	x	1000	=	2000
1	x	10000	=	10000
Total Value			=	12345

 

 

 

 

 

(Recall that 10⁰ is equal to 1.) From this point, we can apply the order of operations and multiply. The first step would give us:

5 + 40 + 300 + 2000 + 10000.

Adding these values together, we get 12345 in standard notation.

Example: Convert 1 x 10⁰ + 9 x 10¹ + 6 x 10² + 2 x 10³ + 7 x 10⁴ + 4 x 10⁵ to standard notation.

Solution: If we follow the rules of order of operations, we must first evaluate the powers of ten.

1 x 1 + 9 x 10 + 6 x 100 + 2 x 1000 + 7 x 10000 + 4 x 100000

Next, we perform the multiplication.

1 + 90 + 600 + 2000 + 70000 + 400000

Finally, we add the numbers together to obtain the sum:

472691

which is the number in standard form.

1.2 ~ Practice Problems

Directions: Turn off your VCR.  Change the following numbers from expanded notation to standard notation.  When you have finished working the problems, check your answers in the back of the book or click here to view the Answer Key. Turn your VCR back on and watch a teacher work the practice problems.

1. 6 x 10⁰ + 8 x 10¹ + 2 x 10²
2. 8 x 10⁰ + 6 x 10¹ + 5 x 10² + 0 x 10³ + 1 x 10⁴
3. 8 x 10⁰ + 6 x 10¹ + 5 x 10² + 1 x 10⁴
4. 3 x 10⁰ + 2 x 10¹
5. 7 x 10⁰
6. 4 x 10⁰ + 7 x 10¹ + 3 x 10² + 6 x 10³ + 8 x 10⁴ + 5 x 10⁵ + 0 x 10⁶ + 1 x 10⁷
7. 4 x 10⁰ + 7 x 10¹ + 3 x 10² + 6 x 10³ + 8 x 10⁴ + 5 x 10⁵ + 1 x 10⁷

 1.2 ~ Homework Problems

Directions: Turn off your VCR.  Convert the following numbers from expanded notation to standard notation.  Keep your work in a notebook to be turned into your instructor at the conclusion of the course.  When you have finished working the problems, check your answers in the back of the book or click here to view the Answer Key. If you have any questions, contact your instructor.

1. 1 x 10⁰ + 3 x 10¹ + 4 x 10² + 7 x 10³
2. 5 x 10⁰ + 2 x 10¹ + 7 x 10³ + 8 x 10⁵
3. 6 x 10¹
4. 5 x 10⁰ + 2 x 10¹ + 4 x 10² + 7 x 10³ + 1 x 10⁴
5. 8 x 10⁰
6. 4 x 10⁰ + 9 x 10¹ + 2 x 10² + 6 x 10³ + 5 x 10⁴ + 1 x 10⁵ + 7 x 10⁶
7. 4 x 10⁰ + 4 x 10¹ + 4 x 10⁴
8. 9 x 10² + 6 x 10³ + 4 x 10⁴ + 8 x 10⁵
9. 3 x 10³ + 9 x 10⁴
10. 7 x 10⁷

This Concludes your work in Session One

Last Updated January 2004