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Scientific Notation
In the sciences, many of the things measured or calculated involve numbers that are either very large or very small. As a result, it is inconvenient to write out such long numbers or to perform calculations long hand. In addition, most calculators do not have enough window space to be able to show these long numbers. To remedy this, we have available a short hand method of representing numbers called scientific notation. The chart below gives you some examples of powers of 10 and their names and equivalences.
Exponent Expanded Prefix Symbol Name Fraction
10 -12^ 0.000000000001 pico- p one trillionth 1/1,000,000,000,
10 -9^ 0.000000001 nano- n one billionth 1/1,000,000,
10 -6^ 0.000001 micro- u one millionth 1/1,000,
10 -3^ 0.001 milli- m one thousandth
10 -2^ 0.01 centi- c one hundredth 1/
10 -1^ 0.1 deci- d one tenth 1/
10 0 1 ------ ------ one --------
10 1 10 Deca- D ten --------
10 2 100 Hecto- H hundred --------
10 3 1,000 Kilo- k thousand --------
10 4 10,000 ------ 10k ten thousand --------
10 5 100,000 ------ 100k one hundredthousand --------
10 6 1,000,000 Mega- M one million --------
10 9 1,000,000,000 Giga- G one billion --------
10 12 1,000,000,000,000 Tera- T one trillion --------
10 15 1,000,000,000,000,000 Peta- P one quadrillion --------
Powers of 10 and Place Value..........
Multiplying by 10, 100, or 1000 in the following problems just means to add the number of zeroes to the number being multiplied. This is because our number system is based on 10. The chart above shows the powers of 10 you are most likely to encounter in your science studies.
- 35 x 10 → 35 + 0 → 350
- 6 x 100 → 6 + 0 + 0 → 600
- 925 x 10 → 925 + 0 → 9,
- 42 x 1000 → 42 + 0 + 0 + 0 → 42,
- 691 x 1000 → 691 + 0 + 0 + 0 → 691,
Places to right of the decimal point are called decimal fractions. The negative exponents shown under the negative exponents shown under the Exponents column above tell you to divide by that number.
Examples:
10 -1^ = 1/10 =. 10 -2^ = 1/10 2 = 1/100 =. 10 -3^ = 1/10 3 = 1/1000 =.
You know that the value of each digit depends on which place it occupies. For example, the 5 in 5628 has the value of 5 thousand, while the value of the 1 in 1586000 is 1 million. A number can be expanded according to the place value it holds:
Example A:
589 = 5(100) + 8(10) + 9(1) 67.32 = 6(10) + 7(1) + 3(.1) + 2(.01)
For larger numbers, it is easier to use exponents for the place values.
Example B:
96,734,000 = 9(10 7 ) + 6(10 6 ) + 7(10 5 ) + 3(10 4 ) + 4(10 3 )
Example Set 2.
Expand each of the following numbers by place (See examples above)
a) 380 _______________________________________
b) 5000.02 _______________________________________
c) 60,400 _______________________________________
d) 29,000,000 _______________________________________
e) 100.004 _______________________________________
Example Set 3.
Write each of the following numbers as a digit times a power of 10. [Ex: 4,000,000 = 4(10 6 )]
a) 50 _____________________________
b) 0.5 _____________________________
c) 80,000 _____________________________
d) 800 _____________________________
e) 0.09 _____________________________
f) 9,000 _____________________________
g) 600,000,000 _____________________________
h) 0.006 _____________________________
i) 30.000 _____________________________
j) 30,000,000 _____________________________
Multiplying and Dividing a Number By a Power of 10
In the last section you saw how trailing zeroes are carried along when you multiply by powers of 10. Adding trailing zeroes on is just like moving the decimal point.
To multiply a number by a power of 10, move the decimal point to the right the same number of places as the exponent.
49 x 100 = 4, 49.00 x 100 = 4,900.
Since you are multiplying by 100 (10 2 ), move the decimal point 2 places to the right. Add zeroes when necessary.
325 x 1,000 = 325, 325.000 x 1000 = 325,
You can multiply by 1000 (10 3 ) by moving the decimal point 3 places to the right. Add zeroes when necessary.
Dividing by powers of 10 can be viewed in the same manner.
To divide a number by a power of 10, move the decimal point to the left the same number of places as the exponent.
6,000 = 60.00 = 60 100
100 can be written as 10^2 , so you would move the decimal point 2 places to the left.
40,000 = 4.0000 = 4 10,
10,000 is the same as 10^4 , so you would move the decimal point 4 places to the left.
To make sense about which way to move the decimal point use the following tips:
- By moving the decimal point to the right, you are making the number larger.
- By moving the decimal point to the left, you are making the number smaller.
D.) 0.00029 in scientific notation is 2.9 x 10 -4.
Move the decimal point so that there is only one non-zero digit to the left. How many places did you move it? In what direction? 4 to the right (multiplied) Compensate for the multiplication by dividing the same number 2.9 x 10 -^.
E. 4.39 x 10^7 written the long way would be 43,900,000.
The single digit 4 is before the decimal point. Add enough zeros so that the 4 is in the 10 7 place. 43,900,000 (The 10 7 place means there are 7 digits between the 4 and the decimal point.)
Example Set 5.
Write the following numbers in scientific notation.
a) 70,000 _____________________________
b) 300,000 _____________________________
c) 800,000,000 _____________________________
d) 9 ,000,000,000 _____________________________
e) 0.008 _____________________________
f) 0.00003 _____________________________
g) 0.000009 _____________________________
h) 0.0000002 _____________________________
Example Set 6.
a) In 1980, major airlines flew 5,400,000 flights. Write the number of flights in scientific notation.
b) In 1986, ($1.25 x 10 10 was spent on state-run lotteries. Write this amount in long form.
c) A pollen grain measures 0.0004 m in diameter. Write this measurement in scientific notation.
d) The radius of the hydrogen atom is 10-8^ cm. Write this in long form.
e) To measure long distances in space, astronomers use a unit called a light-year. A light- year is approximately 5,880,000,000,000 miles long. Write this in scientific notation.
f) Oprah Winfrey's salary as a TV talk show host was reportedly $3.5 x 10 6 for one year. Write this in long form.
g) In 1990 the budget deficit reduction plan was to trim $500 billion from the deficit. Write this figure in scientific notation.
Developed by Gary L. Morrison Student Learning Assistance Center (SLAC) San Antonio College
