## REAL NUMBER SYSTEM

Naturals, Integers, Fractions, Decimals, Rationals, Square Roots, Irrationals, Reals, Formulas, Graphing

Scientific Notation was developed to easily represent numbers that are either very large or very small. Here are two examples of large and small numbers. They are expressed in decimal form instead of scientific notation to help illustrate the problem:

• The mass of the Earth is 5,980,000,000,000,000,000,000,000 kilograms.
• The mass of an alpha particle, which is emitted in the radioactive decay of Plutonium-239, is 0.000,000,000,000,000,000,000,000,006,645 kilograms.

As you can see, it could get tedious writing out those zeros repeatedly, so Scientific Notation was developed to help represent these numbers in a way that was easy to read and understand.

Large Numbers - The mass of the Earth can be written as -

mE = 5.98 x 1,000,000,000,000,000,000,000,000 kg

It is that large number, 1,000,000,000,000,000,000,000,000 which cause the problem. But that is just a multiple of ten. In fact it is ten times itself twenty four times. A more convenient way of writing it is

1,000,000,000,000,000,000,000,000 = 1024.

The small number to the right of the ten is called the "exponent," or the "power of ten." It represents the number of zeros that follow the 1. This enables us to write the mass of the Earth as -

mE = 5.98 x 1024 kg

Small Numbers - The mass of an alpha particle can be written as -

ma = 6.645 x 0.000,000,000,000,000,000,000,000,001 kg

It is that small number, 0.000,000,000,000,000,000,000,000,001 which cause the problem. But that is just a multiple of one tenth. In fact it is one tenth times itself twenty seven times. A more convenient way of writing it is

0.000,000,000,000,000,000,000,000,001 = (1/10)27 = 10-27.

Again the small number to the right of the ten is called the "exponent," or the "power of ten." Note that here it is negative, and it represents the decimal position of the 1. This enables us to write the mass of the alpha particle as -

ma = 6.645 x 10-27 kg

Definition: Any decimal number can be written as a product of a decimal number between 1 and 10 multiplied by an integer power of ten. When this is done the number is written in scientific notation.

Complete the following problems on a separate sheet of paper. Be sure to show all of the work necessary to complete the problems. (Calculators may be used to check the answer, but should not be used to solve the problem originally.)

Numbers Larger than 10

Example:

1. Write 670,620 in scientific notation.

Step 1 - Move the decimal point of 670,620 to the left to get a number between 1 and 10 times a power of ten.

670,620 = 6.7062 x 10exponent

Step 2 - Find the exponent by counting how many places you had to move the decimal point. This is the value of the exponent.

exponent = 5

Step 3 - Write the number in scientific notation.

670,620 = 6.7062 x 105

HOMEWORK PROBLEMS :

2. Write 340.67 in scientific notation.

3. Write 93,000,000 in scientific notation.

4. Write 300,000,000 in scientific notation.

5. Write 1,840,000,000,000,000 in scientific notation.

Numbers smaller than 1

Example:

6. Write 0.00234 in scientific notation.

Step 1 - Move the decimal point of 0.00234 to the right to get a number between 1 and 10 times a power of ten.

0.00234 = 2.34 x 10exponent

Step 2 - Find the exponent by counting how many places you had to move the decimal point. This is the value of the exponent.

exponent = -3

Step 3 - Write the number in scientific notation.

0.00234 = 2.34 x 10-3

HOMEWORK PROBLEMS :

7. Write 0.00005723 in scientific notation.

8. Write 0.00000001032 in scientific notation.

9. Write 0.000000000025789 in scientific notation.

10. Write 0.0000000000000000000097531 in scientific notation.

Addition - The key to adding numbers in Scientific Notation is to make sure the exponents are the same.

Example:

11. Add 6.7062 x 105 plus 3.4067 x 102.

Step 1 - Rewrite both numbers with the same power of 10.

6.7062 x 105 = 6706.2 x 102

3.4067 x 102 = 3.4067 x 102

Step 2 - Add the decimal numbers and multiply by the common power of 10.

6706.2 x 102 + 3.4067 x 102 = 6709.6067 x 102

Step 3 - Write the answer in scientific notation.

6.7062 x 105 + 3.4067 x 102 = 6.7096067 x 105

HOMEWORK PROBLEMS :

12. Add 3 x 108 plus 9.3 x 107.

13. Add 9.3 x 107 plus 6.7062 x 105.

14. Add 6.7845 x 10-6 plus 2.34 x 10-2.

15. Add 8.205 x 102 plus 3.4067 x 10-1.

Subtraction - The key to subtracting numbers in Scientific Notation is the same as for addition.

Example: 16. Subtract 6.7062 x 104 from 3.4067 x 105.

Step 1 - Rewrite both numbers with the same power of 10.

6.7062 x 104 = 6.7062 x 104

3.4067 x 105 = 34.067 x 104

Step 2 - Subtract the decimal numbers and multiply by the common power of 10.

34.067 x 104 - 6.7062 x 104 = 27.3608 x 104

Step 3 - Write the answer in scientific notation.

3.4067 x 105 - 6.7062 x 104 = 2.73608 x 105

HOMEWORK PROBLEMS :

17. Subtract 3 x 108 from 9.3 x 107.

18. Subtract 9.3 x 107 from 6.7062 x 105.

19. Subtract 6.7845 x 10-6 from 3.4067 x 10-5.

20. Subtract 6.7062 x 10-2 from 3.4067 x 10-1.

Multiplication - When multiplying numbers expressed in Scientific Notation, the exponents can simply be added together.

Example:

21. Multiply 6.7062 x 105 times 3.4067 x 102.

Step 1 - Multiply the decimal numbers together and multiply the powers of 10 together.

6.7062 x 105 * 3.4067 x 102 = 6.7062*3.4067 x 105 x 102

Step 2 - Express the product of the decimal numbers in scientific notation.

6.7062*3.4067 x 105 x 102 = 2.284601154 x 101 x 105 x 102

Step 3 - Write the answer in scientific notation by adding up all of the exponents.

6.7062 x 105 * 3.4067 x 102= 2.284601154 x 108

HOMEWORK PROBLEMS :

22. Multiply 6.2 x 104 times 3.40 x 107.

23. Multiply 9.3 x 107 times 2.34 x 102.

24. Multiply 6.843 x 10-3 times 3.4067 x 102.

25. Multiply 5.932 x 10-6 times 8.63 x 10-3.

Division - When dividing numbers expressed in Scientific Notation, the exponent of the denominator is subtracted from the exponent of the numerator.

Example:

26. Divide 6.7062 x 105 by 3.4067 x 102.

Step 1 - Divide the decimal numbers together and divide the powers of 10 together.

(6.7062 x 105 ) / (3.4067 x 102 )= (6.7062 / 3.4067) x (105 / 102 )

Step 2 - Express the quotient of the decimal numbers in scientific notation and subtract the denominator exponent from the numerator exponent.

(6.7062/3.4067) x (105 / 102 ) = 1.968532598 x 100 x 103

Step 3 - Write the answer in scientific notation by adding up all of the remaining exponents.

(6.7062 x 105 ) /( 3.4067 x 102 )= 1.968532598 x 103

HOMEWORK PROBLEMS :

27. Divide 6.2 x 104 by 3.40 x 107.

28. Divide 9.3 x 107 by 2.34 x 102.

29. Divide 6.843 x 10-3 by 3.4067 x 102.

30. Divide 5.932 x 10-6 by 8.63 x 10-3.

Square Root - When finding the square root of a number expressed in Scientific Notation, it is necessary to write the number with an even power of 10, since the exponent of the square root of a number is one-half the exponent of the original number.

Example:

31. Find the square root of 6.7062 x 105.

Step 1 - If necessary, rewrite the number with an even power of 10 by moving the decimal one place to the right, and decreasing the exponent by one.

6.7062 x 105 = 67.062 104

Step 2 - Find the square root of the decimal number and divide the exponent by 2 to get the answer.

HOMEWORK PROBLEMS :

32. Find the square root of 3.40 x 107.

33. Find the square root of 2.34 x 1012.

34. Find the square root of 6.843 x 10-3.

35. Find the square root of 5.932 x 10-6.