## Scientific Notation

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Often in astronomy numbers get too big to be conventionally represented. For example the distance from the Sun to the Andromeda galaxy is about 12,000,000,000,000,000,000 miles. The scientific notation gives us a simple way of simplifying the way these large numbers are represented.

The reverse can also be true, that is a number can be too small to represent easily using standard notation. Scientific notation takes care of these problems.

In the scientific notation all numbers are represented as a number which lies between 1 and 9.99999999.... times a power of ten. For example the number 2,500 can be written as 2.5 x 1000. The last factor is then written as 1000 = 10x10x10 = 103. Thus in scientific notation 2,500 = 2.5 x 103, or "two point five times ten to the power of three". The first number (in this case 2.5) is known as the mantissa, and the power of ten (in this case 3) as the exponent.

1. Large numbers For large numbers the power is always positive, as in the above example. The power is equal to the number of places in the first expression after the first (non zero) digit. 40,000,000 has 7 zeroes after the four, and so is the number 4 x 107. 315,000,000,000 has 11 places after the 3, and so is 3.15 x 1011.
2. Small numbers The same principle can be applied to small numbers, such as 0.00000000045. In this case the power is a negative number, equal to the number of zeroes in the number after the decimal point and before the first non-zero digit, plus one. The above number has 9 zeroes after the '.' and before the '4', and so remembering to add 1 to this, we get 4.5 x 10-10.

#### Try these:

What are the following in scientific notation? (Click the blue cross for the answer.)
1. 6,000,000,000,000,000,000,000,000 X
2. 34,599,800,000 X
3. 12.56 X
4. 0.3456789 X
5. 0.0000000332 X
6. 0.000000000000000000000000000000123456789 X
7. 6 trillion X
What are the following in standard notation?
1. 3 x 105 X
2. 2.44 x 10-7 X
3. 6.02 x 1023 X
4. 6.62 x 10-34 X
5. 1.5 x 1011 X
6. 1.44 x 10-1 X

#### Multiplication

When muliplying numbers in scientific notation, remember the following rules: a) multiply the mantissas, and b) add the powers. Using this rule 4 x 107 times 2 x 104 is equal to (4 times 2) x 10(7+4) = 8 x 1011.

If the new mantissa is now greater than ten, adjust by moving the decimal point to the left, and increase the power by one. For example 3.45 x 109 times 6.7 x 107 = (3.45 times 6.7) x 10(9+7) = 23.115 x 1016 = 2.3115 x 1017.

The rule works equally well if one or both of the powers is negative.

2 x 10-6 times 3 x 10-7 = (2 times 3) x 10(-6+(-7)) = 6 x 10-13.

3 x 1012 times 4 x 10-8 = (3 times 4) x 10(12+(-8)) = 12 x 104 = 1.2 x 105.

#### Division

Division is similar, except you divide the mantissas and subtract the powers. For example, 5 x 108 divided by 2 x 103 = (5 divided by 2) x 10(8-3) = 2.5 x 105

If the new mantissa is now less than one, move the decimal point to the right, and decrease the power by one, as in 3 x 106 divided by 7.8 x 1011 = (3 divided by 7.8) x 10(6-11) = 0.3846 x 10-5 = 3.836 x 10-6

#### Try these:

1. 2 x 109 times 4 x 105 X
2. 6.7 x 1016 times 8.44 x 105 X
3. 55.63 times 4 x 105 (remember to converet the 55.63 to scientific notation first) X
4. 2.55 x 10-7 times 6 x 106 X
5. 2.55 x 10-7 divided by 6 x 106 X
6. 0.000000074 divided by 555,666,777,888 X
7. 5 x 10-5 divided by 5 x 10-9 X

In adding numbers you must first make sure that all the terms have the same power. If they do not then you will need to convert. With terms of the same power, add the mantissas to get the new mantissa by the same power that the original terms had. Thus, 3.00 x 108 + 4.00 x 108 + 3.66 x 108 = (3.00 + 4.00 + 3.66) x 108 = 10.66 x 108, which would then be written as 1.066 x 109. On the other hand the sum 4.22 x106 + 3 x 104 cannot be performed at present since the powers are different. First you must convert one, say the latter, to re-write the sum as 4.22 x106 + 0.03 x 106, which can then be added to get (4.22+ 0.03) x 106 = 4.25 x 106.

Subtraction is the same except you subtract the mantissas.

#### Try these:

1. 2 x 109 + 4 x 109 X
2. 6.7 x 1016 + 8.44 x 1016 X
3. 5.563 x 107 + 4 x 105 X
4. 2.55 x 10-7 + 6 x 10-8 X
5. 2.55 x 10-7 - 6 x 10-8 X
6. 0.000000074 - 0.000000555666777888 X
7. 5 x 10-5 - 5 x 10-9 X
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