Scientific Notation

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This includes Scientific notation which is a way of writing very large or very small numbers. A number is written in scientific notation when a number between 1 and 10 is multiplied by a power of 10. For example, 650,000,000 can be written in scientific notation as 6.5 ✕ 10^8.
1. 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
-12
10 0.000000000001 pico- p one trillionth 1/1,000,000,000,000
-9
10 0.000000001 nano- n one billionth 1/1,000,000,000
-6
10 0.000001 micro- u one millionth 1/1,000,000
one
10-3 0.001 milli- m 1/1,000
thousandth
10-2 0.01 centi- c one hundredth 1/100
-1
10 0.1 deci- d one tenth 1/10
100 1 ------ ------ one --------
101 10 Deca- D ten --------
2
10 100 Hecto- H hundred --------
3
10 1,000 Kilo- k thousand --------
4
10 10,000 ------ 10k ten thousand --------
one hundred
105 100,000 ------ 100k --------
thousand
106 1,000,000 Mega- M one million --------
9
10 1,000,000,000 Giga- G one billion --------
1012 1,000,000,000,000 Tera- T one trillion --------
1015 1,000,000,000,000,000 Peta- P one quadrillion --------
2. 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.
1. 35 x 10 → 35 + 0 → 350
2. 6 x 100 → 6+0+0 → 600
3. 925 x 10 → 925 + 0 → 9,250
4. 42 x 1000 → 42 + 0 + 0 + 0 → 42,000
5. 691 x 1000 → 691 + 0 + 0 + 0 → 691,000
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 = .1
10-2 = 1/102 = 1/100 = .01
10-3 = 1/103 = 1/1000 = .001
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(107) + 6(106) + 7(105) + 3(104) + 4(103)
3. When you expand the numbers you should notice these two facts:
1. When a power of 10 is written the long way, the number of zeroes behind the 1 is the
same as the exponent of the 10 when it is written in shorthand.
Examples:
100 = 102 ; (2 zeroes so exponent is 2)
100,000 = 105 ; (5 zeroes so exponent is 5)
2. The number of places between the digit and the decimal point is the same as the
exponent of the 10 in the place value of that digit. Consider the two following examples:
a. In the number 5,078.4 look at the 5. There are 3 places between it and the decimal
point. Its place value is 103.
b. In the number 56,700,000.6 look at the 5. There are 7 places between it and the
decimal point. Its place value is 107.
On the place value chart below, the place value names do not center around the decimal point.
Instead they center around the ones place. The names to the left of the ones place match up
with the names to the right.
thousands............................................................................thousandths
hundreds..................................................hundredths
tens.................................tenths
ones
Example Set 1.
What is the value of 8 in each of the following numbers?
a) 84.67 _________________________
b) 209.82 _________________________
c) 38,009 _________________________
d) 85,000,000 _________________________
e) 0.08 _________________________
4. 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(106)]
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 _____________________________
5. 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 = 4,900
49.00 x 100 = 4,900.
Since you are multiplying by 100 (102), move the decimal point 2 places to the right. Add
zeroes when necessary .
325 x 1,000 = 325,000
325.000 x 1000 = 325,000
You can multiply by 1000 (103) 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 102 , so you would move the decimal point 2 places to the left.
40,000 = 4.0000 = 4
10,000
10,000 is the same as 104, 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:
1. By moving the decimal point to the right, you are making the number larger.
2. By moving the decimal point to the left, you are making the number smaller.
6. Example Set 4.
Complete the table using the technique of moving the decimal point to multiply and divide.
Number to Use x 10 ÷ 1,000 x 1,000
Ex. 308.5 3,085 0.3085 308,500
Ex. 5.004 50.04 0.005004 5,004
a) 730.2
b) 2.105
c) 45
d) 0.08
Scientific Notation
The population of California in 1990 was estimated at 29,600,000. Another way of writing large
numbers like this, as well as small numbers, is to use a shorthand method called scientific
notation. Thus, the population of California can be written as 2.96 x 107. Scientific notation
consists of two parts: a number with a value between 1 and 10 multiplied by a power of 10.
A.) 5,000,000,000 in scientific notation is 5 x 109.
This is breaking the number into the digit times its place value.
(Count the O's.)
B.) 0.004 in scientific notation would be 4 x 10-3
The 4 is in the thousandths place, so multiplying by 10-3 (1/1000)
is the same as dividing by 103 (1,000).
C.) 8,200 in scientific notation is 8.2 x 103.
Move the decimal point so that there is only one non-zero digit to
the left of it. How many places did you move it? In what direction?
3 to the left (divided). Compensate for the division by multiplying
the same number, 8.2 x 103.
7. 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 10-4.
E. 4.39 x 107 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 107 place. 43,900,000 (The 107 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 _____________________________
8. 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 1010 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 106 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