Scientific Notation is a numbering system that can easily handle the very small or very large
numbers that are sometimes used in scientific investigations. The form of a number expressed
in scientific notation is ‘a’ x 10N, where ‘a’ is a whole number or a decimal between 1 and 10
and N (the exponent) is a whole number.
For numbers greater than 1, the exponent is positive and indicates the number of decimal
places to the right of the decimal point in 'a’.
Examples:
1.2 x 105 = 120,000
4 x 1012 = 4,000,000,000,000
4.65 x 106 = 4,650,000
For numbers less than 1, the exponent is negative and indicates the number of decimal places
to the left of the decimal point in ‘a’.
Examples:
3 x 10-4 = 0.0003
-1
1.0 x 10 = 0.1
1.7 x 10-9 = 0.0000000017
5.74 x 10-6 = 0.00000574
For practice, write the following numbers in decimals:
2.3 x 10-3 =
6.45 x 10-6 =
4.64 x 102 =
1.66 x 104 =
8.9 x 10-7 =
4.3 x 106 =
1.0 x 105 =
7.2 x 10-5 =
Now write the following decimal numbers in scientific notation:
0.0032 =
23,000 =
1,000,000 =
0.45 =
0.0000061 =
71,000,000,000 =
0.01 =
0.0000000009 =
Multiplication:
When two numbers in scientific notation are multiplied, the numbers are multiplied but the
exponents are added together.
Examples:
(2 x 103)(1.5 x 102) = (2 x 1.5) x 10(3+2) = 3 x 105 (check it by multiplying 2,000 x 150)
(2 x 10-3)(1.5 x 102) = (2 x 1.5) x 10(-3 + 2) = 3 x 10-1 (check it: 0.002 x 150 = 0.3)
(2 x 10-3)(1.5 x 10-2) = (2 x 1.5) x 10(-3 + -2) = 3 x 10-5 (check it: 0.002 x 0.015 = 0.00003)
For practice, do the following problems:
(3 x 103)(6 x 102) =
(7 x 10-2)(9 x 104) =
(1.2 x 10-6)(3.0 x 10-2) =
(1.0 x 10-8)(4.67 x 109) =
Division:
When two numbers expressed in scientific notation are divided, the exponent of the divisor is
subtracted from the exponent of the dividend.
(A)
2 x 106
2
------= 1 x 102 or
-- x 10(6-4) = 1 x 102 = 100
2 x 104
2
Check: 2,000,000 / 20,000 = 200 / 2 = 100
(B)
7.2 x 104
--------- = 2.4 x 102
3 x 102
or
7.2
---- x 10(4-2) = 2.4 x 102
3.0
Check: 72,000 / 300 = 240
(C)
7.2 x 104
--------- = 2.4 x 106
3 x 10-2
7.2
---- x 10(4-(-2)) = 2.4 x 106
3.0
Check: 72,000 / 0.03 = 2,4000,000
(D)
7.2 x 10-4
---------- = 2.4 x 10-2
3 x 10-2
7.2
---- x 10(-4-(-2)) = 2.4 x 10-2
3.0
Check: 0.00072 / 0.03 = 0.024
(E)
7.2 x 10-4
---------- = 2.4 x 10-6
3 x 102
7.2
---- x 10(-4-2) = 2.4 x 10-6
3.0
Check: 0.00072 / 300 = 0.0000024
For practice, do the following problems:
1.2 x 10-4
---------- =
3 x 10-3
4.8 x 105
---------- =
3 x 10-7
2.6 x 102
---------- =
2 x 107
1.1 x 10-6
---------- =
1 x 102
Measuring Temperature:
Scientific temperature measurements are made using the Celsius (which used to be called
centigrade) scale, rather than the Fahrenheit scale. Conversions are made using the following
formulas:
O
C = (OF – 32) x (5 / 9)
O
F = (OC x (9 / 5)) + 32
Remember that the temperature that water will freeze is either zero degrees Celsius (0O C) or
32 degrees Fahrenheit (32O F).
Also remember that the temperature that water will boil is either one hundred degrees Celsius
(100O C) or two hundred twelve degrees Fahrenheit (212O F).
You may want to try to remember that normal human core body temperature is either 36.8O C
or 98.2O F.
Practice converting temperatures with the following problems:
145O F = _______ O C
28O C = ________ O F
16O F = _______ O C
72O C = ________ O F
68O F = _______ O C
6O C = ________ O F
212O F = _______ O C
100O C = ________ O F
Measuring Length:
The meter (m) is the basic unit of linear measurement in the metric system. As a comparison
to American Standard units, a meter is a little longer than a yard. Longer distances are
measured in kilometers (km). One kilometer equals 1000 meters and is approximately sixtenths of a mile. The meter is subdivided into centimeters (cm) and millimeters (mm). One
centimeter is one one-hundredth of a meter (1 cm = 0.01 m). One millimeter is one onethousandth of a meter ( 1 mm = 0.001 m). Micrometers (um), which are also called microns,
and nanometers (nm) are smaller subdivisions that are used for measurement of microscopic
objects.
1000.0 meters in a kilometer.
1000 m = 1.0 km
100 centimeters in a meter
100 cm = 1.0 m
1000 millimeters in a meter
1000 mm = 1.0 m
1,000,000 micrometers or microns in a meter
1 x 106 um = 1.0 m
1,000,000,000 nanometers in a meter
1 x 109 nm = 1.0 m
Interconversions: Remember that if two numbers are equal, they can be represented as a
fraction with either number on the top or bottom. For example you can correctly write either
100 cm
--------1.0 m
or
1.0 m
--------100 cm
So how many centimeters in half a meter? Simply set up the ratios so your units cancel.
100 cm
0.5 m x --------- =
1.0 m
(0.5 x 100) cm = 50 cm
How many mm in 50 cm? Again set up your equivalent ratios so the units always cross
cancel.
1.0 m
(50 x 1.0)
50 cm x --------- = ------------ m = 0.5 m But we are not done yet.
100 cm
100
We want mm. So now we simply convert 0.5 m into mm with our ratio:
1000 mm
(0.5 x 1000)
0.5 m x ------------ = ---------------- mm = 500 mm, and so 50 cm = 500 mm.
1.0 m
1.0
Practice using metric units by doing the following conversions:
1500 m = __________ km
350 nm = ____________ um
1.2 cm = ___________ mm
10 km = _____________ m
100 cm = ___________ m
175 mm = ___________ cm
600 nm = ___________ um
175 mm = ___________ m
The length of the human esophagus is 2.4 x 10-1 m or _____________ cm.
The thickness of the plasma membrane of a human cell is 7.0 x 10-9 m or ____________ nm.
The diameter of a typical human cell is 3 x 10-5 m or ____________ um.
The diameter of a human eyeball is 2.5 x 10-2 m or ____________ cm.
Height of an average human female is 1.6 m or _____________ cm.
Length of the small intestine is 600 cm or ____________ m.
Measuring Mass:
The gram (g) is the basic unit for measuring mass in the metric system. Larger amounts are
measured in kilograms (kg). One kilogram equals one thousand grams (1.0 kg = 1000 gms).
As a comparison to American Standard units, a kilogram is approximately 2.2 pounds (2.2
lbs). It takes approximately 28 grams to make an ounce and 454 g to make a pound. Like the
meter, the gram is also divided into smaller units.
If 2.2 pounds = 1.0 kg, how many kilograms do you weight? Ok, I won’t ask that.
Let’s calculate how many kilograms in an average 180 pound man.
1.0 kg
(180 x 1.0)
180 lbs x ---------- = -------------- = 81.8 kg
2.2 lbs
2.2
Practice using metric units by doing the following conversions:
100 kg = ___________ lbs
100 g = ___________ kg
50 lbs = ___________ kg
50 lbs = ___________ g
1500 g = ____________ lbs
0.5 lbs = ____________ g
When measuring masses less than a kilogram, the following units are used. One thousand
milligrams (mg) = 1.0 gram. One million micrograms (ug) equal 1.0 gram. And a
nanogram (ng) is one thousand times lighter than a microgram.
1000 mg = 1.0 g
1.0 g
----------1000 mg
or
1 x 109 ng = 1.0 g
1,000,000 ug = 1.0 g
1 x 106 ug
------------1.0 g
1000 mg
----------1.0 g
1 x 109 ng
-------------1.0 g
Practice using metric mass units by doing the following conversions:
60 mg = ___________ g
2 ug = ____________ ng
240 g = ____________ kg
0.4 kg = ____________ mg
1.7 mg = ____________ ug
3 ng = ______________ ug
Measuring Volume:
The liter (l or L) is the basic unit of volume of fluids in the metric system. As a comparison to
American Standard units, a liter is slightly larger than a quart. The liter is subdivided into
milliliters (ml). One liter = 1000 ml. Milliliters can be further subdivided into microliters
(ul). One milliliter = 1000 ul.
1000 ml = 1.0 liter
1000 ml
1.0 L
--------- or --------1.0 L
1000 ml
and
1,000,000 ul = 1.0 liter
1 x 106 ul = 1.0 liter
1,000,000 ul
--------------1.0 L
or
1.0 L
--------------1,000,000 ul
Practice using metric units by doing the following conversions:
4.0 ml = __________ L
10 L = _____________ ml
300 ml = ___________ ul
37 ml = _____________ L
0.2 L = ____________ ml
750 ul = _____________ ml
It just so happens that one milliliter of water will weight one gram. (1.0 ml = 1.0 g)