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Scientific Notation

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Scientific Notation

In science, measurements are often extremely small or extremely large.  It is inefficient to write the many zeroes in very small numbers like 0.00000000000000523.  Usually, the order of magnitude and the first few digits of the number are what people are interested in.  How should you represent these extreme numbers?

Guidance

Scientific notation is a means of representing very large and very small numbers in a more efficient way.  The general form of scientific notation is $a \cdot 10^b$

The $a$ is a number between 1 and 10 and most often includes a decimal.  The integer $b$ is called the order of magnitude and is a measure of the general size of the number.  If $b$ is negative then the number is small and if $b$ is positive then the number is large.

$1, 240, 000 &= 1.24 \cdot 10^6\\0.0000354 &= 3.54 \cdot 10^{-5}$

Note that when switching to and from scientific notation the sign of $b$ indicates which direction and how many places to move the decimal point.

Multiplying and dividing numbers that are in scientific notation is just an exercise in exponent rules:

$(a \cdot 10^x) \cdot (b \cdot 10^y) &= a \cdot b \cdot 10^{x+y}\\(a \cdot 10^x) \div (b \cdot 10^y) &= \frac{a}{b} \cdot 10^{x-y}$

Addition and subtraction require the numbers to have identical order of magnitudes.

$1.2 \cdot 10^6-5.5 \cdot 10^5=12 \cdot 10^5-5.5 \cdot 10^5=6.5 \cdot 10^5$

Example A

An electron’s mass is about 0.000 000 000 000 000 000 000 000 000 000 910 938 22 kg.

Write this number in scientific notation.

Solution: $9.109 \ 3822 \cdot 10^{-31}$

Example B

The Earth’s circumference is approximately 40,000,000 meters.  What is the radius of the earth in scientific notation?

Solution:  The relationship between circumference and radius is $C=2 \pi r$

$4.0 \cdot 10^7 &= 2 \pi r\\r &= \frac{4.0}{2 \pi} \cdot 10^7 \approx 0.6366 \ldots \cdot 10^7=6.366 \cdot 10^6$

Note that the number of significant digits required depends on the context.

Example C

Simplify the following expression.

$x=(4.56 \cdot 10^7) \cdot (2.89 \cdot 10^8) \div (7.15 \cdot 10^{-15})+216$

Solution: $x=4.56 \cdot 2.89 \cdot 7.15 \cdot 10^0+216=310.22556$

Concept Problem Revisited

In order to represent an extremely large or small number you should count the number of moves necessary for the decimal point to be directly after the first non-zero digit.  This count will be the order of magnitude and will be used as the exponent of 10 as a means of representing how large or small the number is.

Vocabulary

Order of magnitude is formally the exponent in scientific notation.  Informally it refers to size.  Two objects or numbers are of the same order of magnitude are relatively similar sizes.  A marble and a planet are not of the same order of magnitude, but Earth and Venus are.

Scientific notation is a means of representing a number as a product of a number between 1 and 10 and a power of 10.

Guided Practice

1. Order the following numbers from least to greatest.

$5.411 \cdot 10^{-3} \quad 7.837 \cdot 10^{-4} \quad 9.999 \cdot 10^3 \quad 9.5983 \cdot 10^{-7} \quad 8.0984 \cdot 10^3$

2. Compute the following number and use scientific notation.

$2,000,000^3 \cdot 3,000^4$

3. Simplify the following expression.

$(4.713 \cdot 10^7)+(8.985 \cdot 10^5)-(4.987 \cdot 10^2) \cdot (7.3 \cdot 10^{-6}) \div (6.74 \cdot 10^{-9})$

1. First consider the order of magnitude of each number.  Small numbers have negative exponents.  If two numbers have the same order of magnitude, then compare the actual digits.

$9.5983 \cdot 10^{-7} < 7.837 \cdot 10^{-4} < 5.411 \cdot 10^{-3} < 8.0984 \cdot 10^3 < 9.999 \cdot 10^3$

2. First convert each number to scientific notation individually, then process the exponent and multiplication.

$2,000,000^3 \cdot 3,000^4 &= (2 \cdot 10^6)^3 \cdot (3 \cdot 10^3)^4\\&= 8 \cdot 10^{18} \cdot 81 \cdot 10^{12}\\&= 648 \cdot 10^{30}\\&= 6.48 \cdot 10^{32}$

3. Resolve in order of standard order of operations

$(4.713 \cdot 10^7)+(8.985 \cdot 10^5)-(4.987 \cdot 10^2) \cdot (7.3 \cdot 10^{-6}) \div (6.74 \cdot 10^{-9})$

$&= (4.713 \cdot 10^7) + (8.985 \cdot 10^5)-(5.40135 \cdot 10^5)\\&= (471.3 \cdot 10^5)+(8.985 \cdot 10^5)-(5.40135 \cdot 10^5)\\&= 474.8836499 \cdot 10^5\\&= 4.748836499 \cdot 10^7$

Practice

Write the following numbers in scientific notation.

1. 152,780

2. 0.00003256

3. 56, 320

4. 0.0821

5. 1, 000, 000, 000, 000, 000, 000, 000

6. 7.32

7. If the federal budget is \$1.5 trillion, how much does it cost each individual, on average, if there are 300,000,000 people?

8. The Library of Congress has about 60,000,000 items.  How could you express this number in scientific notation?

9. The sun develops $5 \times 10^{23}$ horsepower per second.  How much horsepower is developed in a day? In a year with 365 days?

10. A light-year is about 5,869,713,600 miles.  A spacecraft travels $8.23 \times 10^4$ miles per hour.  How long will it take the spacecraft to travel a light–year?

11. Compute the following number and use scientific notation: $324, 000 \cdot 30,000^3$.

12. Compute the following number and use scientific notation: $14, 300 \cdot 20, 200^2$.

Simplify the following expressions.

13. $(3.29 \cdot 10^4)-(3.295 \cdot 10^5)+(1.25 \cdot 10^2) \cdot (3.97 \cdot 10^{15}) \cdot (5.8 \cdot 10^{-6})$

14. $(1.95 \cdot 10^2)+(6.798 \cdot 10^6)+(2.896 \cdot 10^3) \cdot (5.6 \cdot 10^{-3}) \div (2.89 \cdot 10^4)$

15. $(2.158 \cdot 10^7) \cdot (1.679 \cdot 10^6) - (9.98 \cdot 10^4) \cdot (3.4 \cdot 10^{-2})$