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The and Metric Conversions

Metric Unit Meaning of Metric Equivalent

Units of Length

1 Kilometer (Km) Kilo = 1000 1000 meters (m) or 103 (m)

1 meter (m) Standard

1 centimeter (cm) centi = 1/100th 0.01 meters (m) or 10-2 m

1 millimeter (mm) milli = 1/1000th 0.001 meters (m) or 10-3 m

1 micrometer (μm) micro = 1/1,000,000th 0.000001 meters (m) or 10-6 m

1 nanometer (nm) nano = 1/1,000,000,000th 0.000000001 meters (m) or 10-9 m

Units of

Kilogram (Kg) Kilo = 1000 1000 (g) or 103 (g)

1 (g) Standard Unit of Mass

1 milligram (mg) milli = 1/1000th 0.001 gram (g) or 10-3 g

1 microgram (μg) micro = 1/1,000,000th 0.000001 gram (g) or 10-6 g

1 nanogram (ng) nano = 1/1,000,000,000th 0.000000001 gram (g) or 10-9 g

Units of

Kiloliter (KL) Kilo = 1000 1000 liters (L) or 103 (L)

1 Liter (L) Standard Unit of Volume

1 milliliter (mL) milli = 1/1000th 0.001 Liter (L) or 10-3 L

1 microliter (μL) micro = 1/1,000,000th 0.000001 Liter (L) or 10-6 L

1 nanoliter (nL) nano = 1/1,000,000,000th 0.000000001 Liter (L) or 10-9 L

1 Quantity Measured Units Symbol Relationship meter m 1 m = 39.36 inches centimeter cm 1 m = 100 cm Length milimeter mm 1 m = 1000 mm, or 1 x 103 mm micrometer μm 1 m = 1,000,000 μm, or 1 x 106 μm nanometer nm 1 m = 1,000,000,000 nm, or 1 x 109 nm gram g 1 g = 0.0353 ounces, or 3.53 x 10-2 ounces miligram mg 1 g = 1000 mg, or 1 x 103 mg Mass (Weight)* microgram μg 1 g = 1,000,000 μg, or 1 x 106 μg nanogram ng 1 g = 1,000,000,000 nm, or 1 x 109 ng Kg 1000 g = 1 Kg Time s

Temperature degrees °C Liter L 1 L = 1.06 quarts mililiter mL 1 L = 1000 mL, or 1 x 103 mL Volume Microliter μL 1 L = 1,000,000 μL, or 1 x 106 μL cubic centimeter cc 1 mL = 1 cc, or 1 cc = 1 x 10-3 L

Prefix: Symbol: Magnitude: Meaning (multiply by): Yotta- 1024 1 000 000 000 000 000 000 000 000 Zetta- Z 1021 1 000 000 000 000 000 000 000 Exa- E 1018 1 000 000 000 000 000 000 Peta- P 1015 1 000 000 000 000 000 Tera- T 1012 1 000 000 000 000 Giga- G 109 1 000 000 000 Mega- M 106 1 000 000 kilo- 103 1000 hecto- h 102 100 deka- da 10 10 deci- d 10-1 0.1 centi- c 10-2 0.01 milli- m 10-3 0.001 micro- u () 10-6 0.000 001 nano- n 10-9 0.000 000 001 pico- p 10-12 0.000 000 000 001 femto- f 10-15 0.000 000 000 000 001 atto- a 10-18 0.000 000 000 000 000 001 zepto- z 10-21 0.000 000 000 000 000 000 001 yocto- y 10-24 0.000 000 000 000 000 000 000 001

2 Converting between and normal numbers

• Convert from Scientific Notation to Real Number: 5.14 x 105 = 514000.0 Scientific notation consists of a coefficient (here 5.14) multiplied by 10 raised to an exponent (here 5). To convert to a real number, start with the base and multiply by 5 tens like this: 5.14 x 10 x 10 x 10 x 10 x 10 = 514000.0. Multiplying by tens is easy: one simply moves the point in the base (5.14) 5 places to the right, adding extra zeroes as needed. • Convert from Real Number to Scientific Notation: 0.000345 = 3.45 x 10-4 Here we wish to write the number 0.000345 as a coefficient times 10 raised to an exponent. To convert to scientific notation, start by moving the decimal place in the number until you have a number between 1 and 10; here it is 3.45. The number of places to the left that you had to move the decimal point is the exponent. Here, we had to move the decimal 4 places to the right, so the exponent is -4.

The notation is based on powers of base number 10. The general format looks something like this:

N X 10x where N= number greater than 1 but less than 10, and x = exponent of 10.

Placing numbers in exponential notation has several advantages. 1. For very large numbers and extremely small ones, these numbers can be placed in scientific notation in order to express them in a more concise form. 2. In addition, numbers placed in this notation can be used in a computation with far greater ease. This last advantage was more practical before the advent of calculators and their abundance.

In scientific fields, scientific notation is still used. Let's first discuss how we will express a number greater than 10 in such notational form. Numbers Greater Than 10 1. We first want to locate the decimal and move it either right or left so that there are only one non- zero digit to its left. 2. The resulting placement of the decimal will produce the N part of the standard scientific notational expression. 3. Count the number of places that you had to move the decimal to satisfy step 1 above. 4. If it is to the left as it will be for numbers greater than 10, that number of positions will equal x in the general expression.

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As an example, how do we place the number; 23419 in standard scientific notation?

1. Position the decimal so that there is only one non-zero digit to its left. In this case we end up with; 2.3419. 2. Count the number of positions we had to move the decimal to the left and that will be x. 3. Multiply the results of step 1 and 2 above for the standard form: So we have: 2.3419 X 10 4

How about numbers less than one? We generally follow the same steps except in order to position the decimal with only one non-zero decimal to its left, we will have to move it to the RIGHT. The number of positions that we had to move it to the right will be equal to -x. In other words we will end up with a negative exponent. Negative exponents can be rewritten as values with positive exponents by taking the inversion of the number.

For example: 10-5 can be rewritten as 1/ 105. Here is an example to consider: Express the following number in scientific notation: 0.000436 1. First, we will have to move the decimal to the right in order to satisfy the condition of having one non-zero digit to the left of the decimal. That will give us: 4.36 2. Then we count the number of positions that we had to move it which was 4. That will equal -X or x = -4 And the expression will be 4.36 X 10-4

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What about numbers between 1 and 10? In those numbers we do not need to move the decimal so the exponent will be zero. For example: 7.92 can be rewritten in notational form as: 7.92 X 100 Now it is your turn. Express the following numbers in their equivalent standard notational form: 1. 123,876.3 2. 1,236,840. 3. 4.22 4. 0.000000000000211 5. 0.000238 6. 9.10

One of the advantages of this notation that was mentioned earlier in the lesson was the ability to compute with them in an easier fashion than with actual numerical equivalents. Let's discuss how one would multiply with such notations. The general format for multiplying using scientific notation is as follows: (N X 10x) (M X 10y) = (N) (M) X 10x+y 1. First multiply the N and M numbers together and express an answer. 2. Secondly multiply the exponential parts together by ADDING the exponents together. 3. Finally multiply the two results for your final answer.

For example: (3 X 104) (1 X 102) 1. First 3 X 1=3 2. (104) (102) = 104+2 = 106 3. Finally 3 X 106 for the answer

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Another example: (4 X 103) (2 X 10-4) 1. First 4 X 2 = 8 2. Secondly (103) (10-4) = 103+(-4) = 103-4 = 10-1 3. Finally 8 X 10-1 would be the answer Now you try some examples. Express the product of the following: 1. (3 X 105) (3 X 106) = ? 2. (2 X 107) (3 X 10-9) = ? 3. (4 X 10-6) (4 X 10-4) = ?

Standard Notation If you should get a computed value where the number that appears before the power of ten is greater than 10 or less than 1, then the following rules apply. For example: 165 X 108 would have to have its decimal position adjusted to make it a standard notational form. You would have to move the decimal to the left two positions. In the case where you move the decimal to the left, you would add a positive 1 to the exponent for EACH position that you moved the decimal. So here we would have to move the decimal two positions to the left which means adding 2 to the exponent. That would make the answer 1.65 X 108+2 = 1.65 X 1010 On the other hand if you were required to move the decimal in the answer to the RIGHT in order to express the standard form, you would subtract 1 for each position to the right you repositioned the decimal. So for example in the exponential equivalent .0078 X 105 it is required that the decimal be moved 3 positions to the right. So the answer would be: 7.8 X 10 5-3 = 7.8 X 10 2 Remember to move the decimal to the left ADD 1 to the exponent for each position that it is moved. To move the decimal to the RIGHT subtract 1 to the decimal for each position the decimal is moved to the right.

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