The Big and Small of Science: Scientific Notation

See prior post leading up to this topic here: http://read-about-it.blogspot.com/2010/05/using-todays-in-news-on-asteroids.html.

Imagine your math homework if you had to add two huge (really, "HUGE") numbers together for every problem. Now imagine multiplying those numbers together for the next night's homework. Imagine your math homework if you had to add two TINY (really, "TINY") numbers together for every problem. Now imagine multiplying those teeny tiny numbers together for the next night's homework. Scientists working with very very big numbers, or, very, very small numbers probably felt the way you just imagined yourself feeling. That is, they felt overwhelmed, or, like they'd never finish their homework, until someone came up with scientific notation. (Did you imagine yourself that way or do you like to spend a week doing one night's homework?)

Students today may look at a page of scientific notation homework and balk, but, trust me, the teacher is doing them a favor. Scientific notation makes immense and Lilliputian numbers alike manageable. What do we mean by that? Take a look at the following quote from Gregory L. Curran (and read more at the cited website):

Scientific notation is simply a method for expressing, and working with, very large or very small numbers. It is a short hand method for writing numbers, and an easy method for calculations. Numbers in scientific notation are made up of three parts: the coefficient, the base and the exponent. Observe the example below:

5.67 x 10⁵

This is the scientific notation for the standard number, 567 000. Now look at the number again, with the three parts labeled.
5.67 x 10⁵
coefficient base exponent

In order for a number to be in correct scientific notation, the following conditions must be true:

1. The coefficient must be greater than or equal to 1 and less than 10.
2. The base must be 10.
3. The exponent must show the number of decimal places that the decimal needs to be moved to change the number to standard notation. A negative exponent means that the decimal is moved to the left when changing to standard notation.

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Source: http://www.fordhamprep.org/gcurran/sho/sho/lessons/lesson25.htm
Accessed: 16 September 2010

Students might be thinking, "Great (sarcastically)! Now we have more vocabulary to learn for homework, on top of the math problems." But, the truth is, that vocabulary can help you understand a speedy way to do math (and science) homework. Let's look further. Perhaps some students tried to find out more. Welcome their efforts.

Using more than one source can help build science literacy. For example, one student might find the following quote from Wikipedia:

Scientific notation, also known as standard form or as exponential notation, is a way of writing numbers that accommodates values too large or small to be conveniently written in standard decimal notation. Scientific notation has a number of useful properties and is often favored by scientists, mathematicians and engineers, who work with such numbers.

In scientific notation all numbers are written like this:

a × 10^b

("a times ten to the power of b"), where the exponent b is an integer, and the coefficient a is any real number (but see normalized notation below), called the significand or mantissa (though the term "mantissa" may cause confusion as it can also refer to the fractional part of the common logarithm). If the number is negative then a minus sign precedes a (as in ordinary decimal notation).......

....Any given number can be written in the form of a×10^^b in many ways; for example 350 can be written as 3.5×10² or 35×10¹ or 350×10⁰.

In normalized scientific notation, the exponent b is chosen such that the absolute value of a remains at least one but less than ten 1 ≤ |a| <>Scientific notation also enables simpler order-of-magnitude comparisons. A proton's mass is 0.0000000000000000000000000016726 kg. If this is written as 1.6726×10⁻²⁷ kg, it is easier to compare this mass with that of the electron, given above. The order of magnitude of the ratio of the masses can be obtained by comparing the exponents instead of the more error-prone task of counting the leading zeros. In this case, −27 is larger than −31 and therefore the proton is roughly four orders of magnitude (about 10000 times) more massive than the electron.

Scientific notation also avoids misunderstandings due to regional differences in certain quantifiers, such as billion, which might indicate either 10⁹ or 10¹².

Source: http://en.wikipedia.org/wiki/Scientific_notation Accessed: 16 September 2010.

Compare and contrast the two quotes in a table:

Scientific notation is simply a method for expressing, and working with, very large or very small numbers	Scientific notation, also known as standard form or as exponential notation, is a way of writing numbers that accommodates values too large or small to be conveniently written in standard decimal notation.
It is a short hand method for writing numbers, and an easy method for calculations	Scientific notation also enables simpler order-of-magnitude comparisons. A proton's mass is 0.0000000000000000000000000016726 kg. If this is written as 1.6726×10−27 kg, it is easier to compare this mass with that of the electron, given above. The order of magnitude of the ratio of the masses can be obtained by comparing the exponents instead of the more error-prone task of counting the leading zeros. In this case, −27 is larger than −31 and therefore the proton is roughly four orders of magnitude (about 10000 times) more massive than the electron.
Numbers in scientific notation are made up of three parts: the coefficient, the base and the exponent	("a times ten to the power of b"), where the exponent b is an integer, and the coefficient
Observe the example below:
5.67 x 105	In scientific notation all numbers are written like this: a × 10b
This is the scientific notation for the standard number, 567 000	Any given number can be written in the form of a×10^b in many ways; for example 350 can be written as 3.5×102 or 35×101 or 350×100.
Now look at the number again, with the three parts labeled
5.67 x 105
In order for a number to be in correct scientific notation, the following conditions must be true:
1. The coefficient must be greater than or equal to 1 and less than 10	a is any real number (but see normalized notation below), called the significand or mantissa (though the term "mantissa" may cause confusion as it can also refer to the fractional part of the common logarithm). If the number is negative then a minus sign precedes a (as in ordinary decimal notation). In normalized scientific notation, the exponent b is chosen such that the absolute value of a remains at least one but less than ten (1 ≤ |a| <>
2. The base must be 10
3. The exponent must show the number of decimal places that the decimal needs to be moved to change the number to standard notation
A negative exponent means that the decimal is moved to the left when changing to standard notation
	Scientific notation also avoids misunderstandings due to regional differences in certain quantifiers, such as billion, which might indicate either 109 or 1012.

If students work in pairs or small groups and each is responsible for one source, they can do such comparisons and then share with the whole class what they have discovered. They will be comparing and contrasting, using new vocabulary orally and in writing, and, likely having fun. This activity will enhance science literacy, build vocabulary and help students understand scientific notation so that they can do the math needed more easily.

Here is another quote on scientific notation:

Exponents: Scientific Notation (page 3 of 5)

Sections: Basics, Negative exponents, Scientific notation, Engineering notation, Fractional exponents

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By using exponents, we can reformat numbers. For very large or very small numbers, it is sometimes simpler to use "scientific notation" (so called, because scientists often deal with very large and very small numbers). ^{[ Source: http://www.purplemath.com/modules/exponent3.htm. Accessed 17 September 2010.]}

The format for writing a number in scientific notation is fairly simple: (first digit of the number) followed by (the decimal point) and then (all the rest of the digits of the number), times (10 to an appropriate power). The conversion is fairly simple. ^{[ Source: http://www.purplemath.com/modules/exponent3.htm. Accessed 17 September 2010.]}

• Write 124 in scientific notation.

This is not a very large number, but it will work nicely for an example. To convert this to scientific notation, I first write "1.24". This is not the same number, but (1.24)(100) = 124 is, and 100 = 10². Then, in scientific notation, 124 is written as 1.24 × 10². ^{[ Source: http://www.purplemath.com/modules/exponent3.htm. Accessed 17 September 2010.]}

Actually, converting between "regular" notation and scientific notation is even simpler than I just showed, because all you really need to do is count decimal places.

^{[ Source: http://www.purplemath.com/modules/exponent3.htm. Accessed 17 September 2010.]}

• Write in decimal notation: 3.6 × 10¹²

Since the exponent on 10 is positive, I know they are looking for a LARGE number, so I'll need to move the decimal point to the right, in order to make the number LARGER. Since the exponent on 10 is "12", I'll need to move the decimal point twelve places over. First, I'll move the decimal point twelve places over. I make little loops when I count off the places, to keep track:

[See image at: http://www.purplemath.com/modules/exponent3.htm]

Then I fill in the loops with zeroes: Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved
[See image at: http://www.purplemath.com/modules/exponent3.htm]

In other words, the number is 3,600,000,000,000, or 3.6 trillion ^{[ Source: http://www.purplemath.com/modules/exponent3.htm. Accessed 17 September 2010.]}

Idiomatic note: "Trillion" means a thousand billion — that is, a thousand thousand million — in American parlance; the British-English term for the American "billion" would be "a milliard", so the American "trillion" (above) would be a British "thousand milliard".

^{[ Source: http://www.purplemath.com/modules/exponent3.htm. Accessed 17 September 2010.]}

• Write 0.000 000 000 043 6 in scientific notation.

In scientific notation, the number part (as opposed to the ten-to-a-power part) will be "4.36". So I will count how many places the decimal point has to move to get from where it is now to where it needs to be:



[See image at: http://www.purplemath.com/modules/exponent3.htm]

Then the power on 10 has to be –11: "eleven", because that's how many places the decimal point needs to be moved, and "negative", because I'm dealing with a SMALL number. So, in scientific notation, the number is written as 4.36 × 10^{–11 [ Source: http://www.purplemath.com/modules/exponent3.htm. Accessed 17 September 2010.]}

.

• Convert 4.2 × 10^–7 to decimal notation.

Since the exponent on 10 is negative, I am looking for a small number. Since the exponent is a seven, I will be moving the decimal point seven places. Since I need to move the point to get a small number, I'll be moving it to the left. The answer is 0.000 000 42 ^{[ Source: http://www.purplemath.com/modules/exponent3.htm. Accessed 17 September 2010.]}

• Convert 0.000 000 005 78 to scientific notation.

This is a small number, so the exponent on 10 will be negative. The first "interesting" digit in this number is the 5, so that's where the decimal point will need to go. To get from where it is to right after the 5, the decimal point will need to move nine places to the right. Then the power on 10 will be a negative 9, and the answer is 5.78 × 10^–9

^{[ Source: http://www.purplemath.com/modules/exponent3.htm. Accessed 17 September 2010.]}

• Convert 93,000,000 to scientific notation.

This is a large number, so the exponent on 10 will be positive. The first "interesting" digit in this number is the leading 9, so that's where the decimal point will need to go. To get from where it is to right after the 9, the decimal point will need to move seven places to the left. Then the power on 10 will be a positive 7, and the answer is 9.3 × 10⁷

^{[ Source: http://www.purplemath.com/modules/exponent3.htm. Accessed 17 September 2010.]}

Just remember: However many spaces you moved the decimal, that's the power on 10. If you have a small number (smaller than 1, in absolute value), then the power is negative; if it's a large number (bigger than 1, in absolute value), then the exponent is positive.

^{[ Source: http://www.purplemath.com/modules/exponent3.htm. Accessed 17 September 2010.]}

Warning: A negative on an exponent and a negative on a number mean two very different things! For instance:

–0.00036 = –3.6 × 10^–4
0.00036 = 3.6 × 10^–4
36,000 = 3.6 × 10⁴
–36,000 = –3.6 × 10⁴

Don't confuse these!

Source: http://www.purplemath.com/modules/exponent3.htm. Accessed 17 September 2010.

Can the students place information in this quote in another column, adding to the information in the two columns in the table above and create a new comparison table? Is some of the information the same? Is there new information?



(c)2010 J S Shipman