MAT 51 Wladis Project 2: Scientific Notation Definition: Scientific notation is any number that is written in a form that is: 
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A decimal with a non‐zero digit to the left of the decimal point in the one’s place, and no other digits to the left of the decimal Multiplied by Ten to some positive or negative exponent So these numbers are written in scientific notation: 
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2.34 10 5.08 10 1.000005 10 5 10 But these are NOT: 2.03 23.4 10 . 13 10 0.13 10 0.00000023 30.1
10 Why not? There is no multiplication by 10 to some power. There is a digit in the ten’s place, as well as the one’s place. There is no digit to the left of the decimal. The digit in the one’s place is zero. There is no multiplication by 10 to some power. There is a digit in the ten’s place. 1) Which of the following numbers are written in scientific notation? Number In scientific notation? (yes/no) 2.4 10 24.1 10 . 241 10 0.24 10 0.00002 2.005 10 0.01 10 20.1 10 If no, why not? Switching between scientific and standard notation: There are two pieces of information to keep in mind: 
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How many places must the decimal move? Very “big” numbers correspond to a positive exponent, whereas very “small” numbers correspond to a negative exponent (because negative exponents denote “big” numbers at the bottom of a fraction, which are actually small decimals) Original number 0.00000000506 2,000,000 5.009
2.1
10 10 How many places should the decimal move? Decimal should be after the 5must move 9 places Decimal should be after the 2must move 6 places Decimal must move 12 places Decimal must move 7 places Big↔positive exponent Number in new notation Small↔negative exponent Smallnegative exponent 5.06 10 2
Bigpositive exponent 10 5,009,000,000,000
Positive exponentBig 0.00000021
Negative exponentSmall 2) Now you try! For the numbers below in standard notation, put them into scientific notation, and vice versa. Model these problems on the ones in the chart above: Original number 0.0000008 8.07 10 40,600,000,000 9.4 10 How many places should the decimal move? Big↔positive exponent Small↔negative exponent Number in new notation Converting numbers that are not quite in scientific notation into scientific notation: Sometimes we might perform some calculations and end up with numbers that are not quite in scientific notation. But it is easy to fix this. For example, none of the following four numbers are in scientific notation because there are either extra digits to the left of the decimal (e.g. digits in the tens place) or because there is no non‐zero digit to the left of the decimal: 
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200.1 10 , 31 10 , 0.7 10 , 0.0201 10 For each of these numbers, we can correctly put them into scientific notation by first putting the “number part” into scientific notation, and then simplifying the two powers with base 10, like this: Original number 200.1 10 31
10
Put “number” part into Group the powers with Simplify the powers with base 10Number is now in scientific notation scientific notation base 10 together 10 ⋅ 10 10 ⋅ 10 ⋅ 10 ⋅ 10 ⋅ 10 ⋅ 10 ⋅ 10
2.001 10
10 2.001
10
10 2.001
2.001 10
3.1 10
10 3.1
10
10
3.1
3.1
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3.1
3.1
0.7
10 7
10
10 7
10
10
2.01
10
10 2.01
10
10
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10
7
7
10 ⋅
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7
0.0201
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10
2.01
2.01
2.01
2.01
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10
3) Now you try! Put each of the following numbers into scientific notation. Model these problems on the ones in the chart above: Original number 406 10
Put “number” part into Group the powers with Simplify the powers with base 10Number is now in scientific notation base 10 together scientific notation 0.05 10 0.87 10 77 10 Performing calculations with numbers in scientific notation is just like other exponent problems, except that your final answer needs to be written in scientific notation form. For example, structurally, these two problems are the same: Simplify, writing your answer using negative exponents: 4
8
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Simplify, writing your answer in scientific notation: 4 10
8 10
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or 0.5
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10 or 0.5
10 Extra steps: 0.5 10 is not in scientific notation, so we need to put it into that form, just like the problems in the last section: 0.5 10 5 10
10 5
10
10 5
5
5
5
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10
In both these types of problems above, the method is the same. The only difference is that in scientific notation problems, we 1) write the final coefficient as a decimal instead of a fraction (For a review of how to divide fractions in order to turn them into decimals, see XXXXXXX.), and 2) we have to rewrite the final answer in scientific notation form if it is not in that form already. 4) Now you try! For each of the following four problems, simplify the exponent problem and the scientific notation problem side‐by‐side, just like we did above. Line up similar steps in each problem, just as we did above. You will need to write your work on another sheet of paper to make room for these problems. For each of the problems, when we say “write your answer using negative exponents”, this is only intended to mean that you should use negative exponents when possible. So, for example, 3 can’t be written in a straightforward way with negative exponents, but should be rewritten as 3
. Simplify, writing your answer using negative exponents: 6
8
Simplify, writing your answer using negative exponents: 4
5
Simplify, writing your answer using negative exponents: 2
3
6
Simplify, writing your answer using negative exponents: 1.5
8 Simplify, writing your answer in scientific notation: 6 10
8 10
Simplify, writing your answer in scientific notation: 5 10 4 10 Simplify, writing your answer in scientific notation: 2 10 3 10
6 10
Simplify, writing your answer in scientific notation: 1.5 10 8 10