BIOMath Essentials
A) Scientific notation
Scientists constantly work with extremely large and small numbers. In standard
notation, numbers such as 38,000,000,000,000,000 km, which is the distance to the star
Centauri or 0.0000000000038 g, which is the mass of a mitochondrion, are awkward
and confusing. Scientific notation provides scientists with a numbering system that is
much easier to use and interpret. A scientific notation is made of three parts: the
coefficient, the base and the exponent.
Let’s take an example to explain how to write in scientific notation:
Standard notation: 93,000,000 Scientific notation: 9.3 x 107
Standard notation: 0.00000093 Scientific notation: 9.3 x 10-7
1. The coefficient is a number greater than 1 but less than 10; it is 9.3 in this
example.
2. The base is always 10.
3. The exponent reflects the number of places the decimal has to be moved to
change the number back to its standard notation. Note that numbers greater than
1 have positive exponents, while numbers less than 1 have negative exponents.
Now, let’s review how to perform basic calculation with exponents:
To multiply, simply multiply the coefficients and add the exponents
o For example,
(3 x 105) x (7 x 103) = (3 x 7) x 10(5+3) = 21 x 108 = 2.1 x 109
(3 x 105) x (7 x 10-3) = (3 x 7) x 10(5+(-3)) = 21 x 102 = 2.1 x 103
(3 x 10-5) x (7 x 10-3) = (3 x 7) x 10((-5)+(-3)) = 21 x 10-8 = 2.1 x 10-9
To divide, simply divide the coefficients and subtract the exponents
o For example,
(4 x 104) ÷ (8 x 102) = (4 ÷ 8) x 10(4-2) = 0.5 x 102 = 5 x 101
(4 x 104) ÷ (8 x 10-2) = (4 ÷ 8) x 10(4-(-2)) = 0.5 x 106 = 5 x 105
(4 x 10-4) ÷ (8 x 102) = (4 ÷ 8) x 10((-4)-2) = 0.5 x 10-6 = 5 x 10-7
(4 x 10-4) ÷ (8 x 10-2) = (4 ÷ 8) x 10((-4)-(-2)) = 0.5 x 10-2 = 5 x 10-3
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, To add, convert both numbers to the same exponent (see unit conversion below),
then add the coefficients, leaving the exponents the same.
o For example,
(2 x 107) + (9 x 106) = (20 x 106) + (9 x 106) = 29 x 106 = 2.9 x 107
(2 x 107) + (9 x 106) = (2 x 107) + (0.9 x 107) = 2.9 x 107
(2 x 10-7) + (9 x 106) = 9 x 106
(2 x 10-7) + (9 x 10-6) = (0.2 x 10-6) + (9 x 10-6) = 9.2 x 10-6
To subtract, convert both numbers to the same exponent, then subtract the
coefficients and leave the exponents unchanged.
o For example,
(5 x 103) - (7 x 104) = (0.5 x 104) - (7 x 104) = - 6.5 x 104
(5 x 103) - (7 x 104) = (5 x 103) - (70 x 103) = - 65 x 103 = - 6.5 x 104
(5 x 103) - (7 x 10-4) = 5 x 103
(5 x 10-3) - (7 x 10-4) = (5 x 10-3) - (0.7 x 10-3) = 4.3 x 10-3
Note: Make sure you can easily perform these calculations without using a calculator.
Go to the Practice File posted on blackboard in Lab # 1 folder to further practice this
skill.
B) Rounding numbers
From: Fact Monster/Information Please® Database, © 2008 Pearson Education, Inc.
A rounded number has about the same value as the number you started with, but it is
less exact.
For example,
341 rounded to the nearest hundred is 300. That is because 341 is closer in value to 300 than to
400.
When rounding off to the nearest dollar, $1.89 becomes $2.00, because $1.89 is closer to $2.00
than to $1.00
Rules for Rounding
Here's the general rule for rounding:
If the number you are rounding is followed by 6, 7, 8, or 9, round the
number up. Example: 38 rounded to the nearest ten is 40
If the number you are rounding is followed by 0, 1, 2, 3, or 4, round the
number down. Example: 33 rounded to the nearest ten is 30
Rounding numbers that ends with 5
In this case, you need to round 5 to the nearest even number.
As a result, about half of the time 5 the number will be rounded up, and about
half of the time it will be rounded down.
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A) Scientific notation
Scientists constantly work with extremely large and small numbers. In standard
notation, numbers such as 38,000,000,000,000,000 km, which is the distance to the star
Centauri or 0.0000000000038 g, which is the mass of a mitochondrion, are awkward
and confusing. Scientific notation provides scientists with a numbering system that is
much easier to use and interpret. A scientific notation is made of three parts: the
coefficient, the base and the exponent.
Let’s take an example to explain how to write in scientific notation:
Standard notation: 93,000,000 Scientific notation: 9.3 x 107
Standard notation: 0.00000093 Scientific notation: 9.3 x 10-7
1. The coefficient is a number greater than 1 but less than 10; it is 9.3 in this
example.
2. The base is always 10.
3. The exponent reflects the number of places the decimal has to be moved to
change the number back to its standard notation. Note that numbers greater than
1 have positive exponents, while numbers less than 1 have negative exponents.
Now, let’s review how to perform basic calculation with exponents:
To multiply, simply multiply the coefficients and add the exponents
o For example,
(3 x 105) x (7 x 103) = (3 x 7) x 10(5+3) = 21 x 108 = 2.1 x 109
(3 x 105) x (7 x 10-3) = (3 x 7) x 10(5+(-3)) = 21 x 102 = 2.1 x 103
(3 x 10-5) x (7 x 10-3) = (3 x 7) x 10((-5)+(-3)) = 21 x 10-8 = 2.1 x 10-9
To divide, simply divide the coefficients and subtract the exponents
o For example,
(4 x 104) ÷ (8 x 102) = (4 ÷ 8) x 10(4-2) = 0.5 x 102 = 5 x 101
(4 x 104) ÷ (8 x 10-2) = (4 ÷ 8) x 10(4-(-2)) = 0.5 x 106 = 5 x 105
(4 x 10-4) ÷ (8 x 102) = (4 ÷ 8) x 10((-4)-2) = 0.5 x 10-6 = 5 x 10-7
(4 x 10-4) ÷ (8 x 10-2) = (4 ÷ 8) x 10((-4)-(-2)) = 0.5 x 10-2 = 5 x 10-3
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, To add, convert both numbers to the same exponent (see unit conversion below),
then add the coefficients, leaving the exponents the same.
o For example,
(2 x 107) + (9 x 106) = (20 x 106) + (9 x 106) = 29 x 106 = 2.9 x 107
(2 x 107) + (9 x 106) = (2 x 107) + (0.9 x 107) = 2.9 x 107
(2 x 10-7) + (9 x 106) = 9 x 106
(2 x 10-7) + (9 x 10-6) = (0.2 x 10-6) + (9 x 10-6) = 9.2 x 10-6
To subtract, convert both numbers to the same exponent, then subtract the
coefficients and leave the exponents unchanged.
o For example,
(5 x 103) - (7 x 104) = (0.5 x 104) - (7 x 104) = - 6.5 x 104
(5 x 103) - (7 x 104) = (5 x 103) - (70 x 103) = - 65 x 103 = - 6.5 x 104
(5 x 103) - (7 x 10-4) = 5 x 103
(5 x 10-3) - (7 x 10-4) = (5 x 10-3) - (0.7 x 10-3) = 4.3 x 10-3
Note: Make sure you can easily perform these calculations without using a calculator.
Go to the Practice File posted on blackboard in Lab # 1 folder to further practice this
skill.
B) Rounding numbers
From: Fact Monster/Information Please® Database, © 2008 Pearson Education, Inc.
A rounded number has about the same value as the number you started with, but it is
less exact.
For example,
341 rounded to the nearest hundred is 300. That is because 341 is closer in value to 300 than to
400.
When rounding off to the nearest dollar, $1.89 becomes $2.00, because $1.89 is closer to $2.00
than to $1.00
Rules for Rounding
Here's the general rule for rounding:
If the number you are rounding is followed by 6, 7, 8, or 9, round the
number up. Example: 38 rounded to the nearest ten is 40
If the number you are rounding is followed by 0, 1, 2, 3, or 4, round the
number down. Example: 33 rounded to the nearest ten is 30
Rounding numbers that ends with 5
In this case, you need to round 5 to the nearest even number.
As a result, about half of the time 5 the number will be rounded up, and about
half of the time it will be rounded down.
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