15
II. Scientific Notation
A. Using Scientific Notation
In science, we have the opportunity to work with both extremely large and extremely
small numbers. However, numbers like this are often extremely cumbersome to write:
602,000,000,000,000,000,000,000
0.000000000000000000000000000911
Scientific notation is a way of writing numbers that is especially useful for very large or
very small numbers. In the examples above, the information about the number’s specific value is
contained in the 602 and the 911, while the rest of the zeros only indicate the placement of the
decimal point, and do not count as significant figures (they are either trailing zeroes or leading
zeroes, respectively). In scientific notation, these numbers can be written more compactly as:
602,000,000,000,000,000,000,000
6.02×1023
0.000000000000000000000000000911
9.11×10-28
Scientific notation is based on the idea that multiplication or division by 10 does not
change the value of the number, only the position of the decimal point:
1.
Multiplying a number by 10 moves the decimal point one place to the right every time
you multiply by 10:
7.61 × 10 = 76.1
2.
0.0053 × 10 = 0.053
Dividing a number by 10 moves the decimal point one place to the left every time you
divide by 10:
85.21
= 8.521
10
3.
4.382 × 10 × 10 = 438.2
0.00633
= 0.000633
10
9163
= 9.163
10 × 10 × 10
Multiplying by 10 several times can be indicated by using positive exponents on the 10:
6.132 × 10 × 10 × 10 = 6.132×103
4.
Dividing by 10 several times can be indicated by using negative exponents on the 10:
4.554
= 4.554 × 10 −2
10 × 10
In scientific notation, we indicate the specific digit values with a number between 1 and
10, followed by a 10 raised to some exponent which indicates how many times the specific digit
values must be multiplied or divided by 10 to obtain the desired decimal number:
16
(1-10)×10exponent
(The meaning of several exponents is listed in Table 1.) For example, the number
602,000,000,000,000,000,000,000 is written as 6.02×1023, since multiplying 6.02 by 10 twentythree times produces the original number. Likewise, 0.000000000000000000000000000911 is
written 9.11×10-28, since 9.11 must be divided by 10 twenty-eight times to produce the original
number. Notice that 52.47×108 is not in scientific notation, because the digit information 52.47
is not between 1 and 10. It must be changed to 5.247×109 to be in proper scientific notation.
Table 1. Exponents in Scientific Notation
6
10
105
104
103
102
101
100
10-1
=
=
=
=
=
=
=
=
10-2 =
-3
=
-4
=
10
10
(10)(10)(10)(10)(10)(10)
(10)(10)(10)(10)(10)
(10)(10)(10)(10)
(10)(10)(10)
(10)(10)
(10)
(1)
(101 ) = (0.1)
(101 ) (101 ) = (0.1)(0.1)
(101 ) (101 ) (101 ) = (0.1)(0.1)(0.1)
(101 ) (101 ) (101 ) (101 ) = (0.1)(0.1)(0.1)(0.1)
=
=
=
=
=
=
=
=
1,000,000
100,000
10,000
1,000
100
10
1
0.1
= 0.01
= 0.001
= 0.0001
Thus, to write a number in scientific notation, take the original number and move its
decimal point either left or right to produce a number between 1 and 10. Then use a power of 10,
either + or -, to indicate how many places the decimal point was moved. If the decimal point
was moved to the right, make the exponent (-); if the decimal point was moved to the left, make
the exponent (+). If there is already a power of 10, moving the decimal to the right makes the
exponent more negative by the number of decimal places moved, while moving the decimal to
the left makes the exponent more positive. The following examples illustrate these rules.
Original No.
0.00512
647
583.71×106
0.0007688×10-3
0.00531×102
Scientific No.
5.12×10-3
6.47×102
5.8371×108
7.688×10-7
5.31×10-1
Comments
Decimal point moved three places to right.
Decimal point moved two places to left.
Decimal to left, 6 becomes more positive by 2.
Decimal to left, -3 becomes more negative by 4.
Decimal to right, 2 becomes more negative by 3.
Negative exponents are sometimes used on units or symbols as well as on numbers. Just
as 1/10 can be written as 10-1, likewise 1/x can be written as x-1, or 1/x3 as x-3. In the same way,
g/mL can be written as g·mL-1 or cm/sec2 as cm·sec-2.
Numbers like 102 or 10-4 should be considered as abbreviations for 1×102 or 1×10-4. Also
note that a number such as 5.73 can be written as 5.73×100, since 100 equals 1.
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B. Arithmetic Using Scientific Notation
Significant Digits. One advantage of scientific notation is that all significant digit
information is contained in the 1-10 portion of the number. For example, 6.5×10-17 contains two
significant digits, and 4.8113×107 contains five significant digits. Even the ambiguous cases can
be clarified, such as the number 5000: to indicate one significant digit, you would write 5×103,
to indicate two significant digits you would write 5.0×103, to indicate three significant digits you
would write 5.00×103, and so on.
Multiplication. All arithmetical operations on numbers written in scientific notation are
performed in two separate parts, one on the 1-10 segments and one on the exponents of 10.
Note: Calculators that do exponential arithmetic perform both parts simultaneously. To multiply
two exponential numbers, first multiply the two 1-10 parts. Then obtain the new power of 10 by
adding the two original powers of 10.
(3.7×102)(2.21×103) = 8.2×105
(1.33×10-12)(6.412×104) = 8.53×10-8
(5.71×10-6)(4.23×10-7) = 2.42×10-12
The last example illustrates a common error: it is easy to look at the exponents -6 and -7, and
expect a final exponent of -13, forgetting that (5.71)(4.23) = 24.2, not 2.42. Thus 24.2×10-13
must be written in correct scientific notation as 2.42×10-12.
Division. To divide one exponential number by another, first divide the two 1-10 parts in
the proper order. Then obtain the new power of 10 by subtracting the two original powers of 10
(exponents) in the order top minus bottom.
7.143 × 10 8
= 2.80 × 10 8−3 = 2.80 × 10 5
3
2.55 × 10
5.6 × 10 9
= 1.4 × 10 9−( −3) = 1.4 × 1012
3.89 × 10 −3
2.971 × 10 −4
= 0.678 × 10 − 4−( −7 ) = 0.678 × 10 3 = 6.78 × 10 2
−7
4.38 × 10
Addition/Subtraction. To add or subtract two or more exponential numbers, each
number must first be written so that all have the same power of 10 (usually the most positive
exponent), even though some may not be in exponential form. Then the parts between 1-10 are
added or subtracted, and the new power of 10 is just the power of 10 to which all numbers were
changed. Answers are rounded to proper number of significant digits as indicated by vertical
dotted line.
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6.132×103
+ 2.88×104
1.66×10-3
– 8.34×10-4
→
→
→
→
0.6132×104
2.88×104
3.4932×104
→
3.49×104
1.66×10-3
0.834×10-3
0.826×10-3
→
0.83×10-3 ═ 8.3×10-4
Raising to a Power. To raise an exponential number to a power, first raise the 1-10 part
to that power. Then to get the new power of 10, multiply the original exponent of 10 by the
power of which the number is being raised.
(1.6×105)3 = (1.6)3×10(5×3) = 4.1×1015
(5.72×10-9)4 = (5.72)4×10(-9)(4) = 1070.49×10-36 = 1.07×10-33
Extracting a Root. To take the root of an exponential number, first take that root of the
1-10 part. Then to get the new exponent of 10, divide the original exponent by the root being
taken. If the original exponent is not initially exactly divisible by the root, first re-write the
entire exponential number, moving the decimal to the right enough places so that the new
exponent of 10 is exactly divisible by the desired root.
6.32 × 10 8 = 6.32 × 10 (8 / 2) = 2.51 × 10 4
3
1.92 × 1013 = 3 19.2 × 1012 = 3 19.2 × 10 (12 / 3) = 2.68 × 10 4
Scientific Notation on Calculators. Most calculators that do any scientific or higherlevel math functions allow a number to be entered in scientific notation. Texas Instrument
calculators usually have a button marked “EE” [or on some newer calculators as “×10x”], which
allows the number 6.02×1023 to be entered as “6.02” + “EE” + “23” (on the screen, it will look
like 6.02E23). Casio and Sharp calculators use a button marked “EXP” to accomplish the same
thing; HP calculators use a button marked “E.” [Note: Do not confuse those buttons with the
“ex”, “exp(x)”, or “10x” buttons, which are logarithmic functions, and are usually paired on the
same button with the “log” and “ln” functions.]