Answer:
hen working with very large or very small numbers, scientists, mathematicians, and engineers often use scientific notation to express those quantities. Scientific notation uses exponential notation. The following are examples of scientific notation.
Light year: number of miles light travels in one year, about 5,880,000,000,000
Scientific notation is 5.88 x 1012 miles.
hydrogen atom: has a diameter of about 0.00000005 mm
Scientific notation is 5 x 10-8 mm
Computation with very large numbers is made easier with scientific notation.
Learning to Use Scientific Notation
When a number is written in scientific notation, the exponent tells you if the term is a large or a small number. A positive exponent indicates a large number and a negative exponent indicates a small number that is between 0 and 1.
Since it’s so useful, let’s look more closely at the details of scientific notation format.
Scientific Notation
A positive number is written in scientific notation if it is written as a x 10n where the coefficient a has a value such that 1 ≤ a < 10 and n is an integer.
Look at the numbers below. Which of the numbers is written in scientific notation?
Number
Scientific Notation?
Explanation
1.85 x 10-2
yes
1 ≤1.85 < 10
-2 is an integer
no
is not an integer
0.82 x 1014
no
0.82 is not ≥ 1
10 x 103
no
10 is not < 10
Which number below is written in scientific notation?
A) 4.25 x 100.08
B) 0.425 x 107
C) 42.5 x 105
D) 4.25 x 106
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Writing Decimal Notation in Scientific Notation
Now let’s compare some numbers expressed in both scientific notation and standard decimal notation in order to understand how to convert from one form to the other. Take a look at the tables below. Pay close attention to the exponent in the scientific notation and the position of the decimal point in the decimal notation.
Large Numbers
Small Numbers
Decimal Notation
Scientific Notation
Decimal Notation
Scientific Notation
500.0
5 x 102
0.05
5 x 10-2
80,000.0
8 x 104
0.0008
8 x 10-4
43,000,000.0
4.3 x 107
0.00000043
4.3 x 10-7
62,500,000,000.0
6.25 x 1010
0.000000000625
6.25 x 10-10
To write a large number in scientific notation, move the decimal point to the left to obtain a number between 1 and 10. Since moving the decimal point changes the value, you have to multiply the decimal by a power of 10 so that the expression has the same value.
Let’s look at an example.
180,000. = 18,000.0 x 101
1,800.00 x 102
180.000 x 103
18.0000 x 104
1.80000 x 105
180,000 = 1.8 x 105
Notice that the decimal point was moved 5 places to the left, and the exponent is 5.
The world population is estimated to be about 6,800,000,000 people. Which answer expresses this number in scientific notation?
A) 7 x 109
B) 0.68 x 1010
C) 6.8 x 109
D) 68 x 108
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Advanced Question
Represent 1.00357 x 10-6 in decimal form.
A) 1.00357000000
B) 0.000100357
C) 0.000001357
D) 0.00000100357
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To write a small number (between 0 and 1) in scientific notation, you move the decimal to the right and the exponent will have to be negative.
0.00004 = 00.0004 x 10-1
000.004 x 10-2
0000.04 x 10-3
00000.4 x 10-4
000004. x 10-5
0.00004 = 4 x 10-5
0.82 x 10-6
= 11.36 moles, being 44
the molar mass of CO₂