Scientific Notation

Most people are accustomed to seeing numbers written in a fairly simple, straightforward form. For example, consider the following numbers.

1,230      8      3.14      1,000,000      144

Such notation works fine in many cases, but if one has to write rather large or small numbers, we adopt various other strategies for expressing the numbers, as follows.

Two dozen       Five million       A gross       Two thousand       One thousandth

Once again, these methods work fine for "even" quantities like "two dozen" or "five million", but they don't work very well for very small numbers like "0.00000246", for fractional quantities like 3.145 thousand, or for very large numbers, particularly large numbers that have several non-zero digits, like 45,648,000,000,000,000,000. To deal with these problems, scientists have adopted various strategies, two of which are generally very useful, and which we will find useful in this astronomy class where we are going to have to use and calculate with some very large and some very small numbers, some of which are known with great precision.

Consider quantities like the number of meters in a light year, which is 9,460,530,000,000,000, the velocity of light in meters per second, which is 299,792,500, or the distance to the Andromeda galaxy, which is written in everyday notation as 21,800,000,000,000,000,000,000 m. All three of these numbers, and many, many others like them are laborious to write out and read in standard, everyday format, and they are even more difficult to calculate with. However, there is a simpler way to express such numbers.

Take the first number given just above; this number would be written in scientific notation as: 9.46053 x 10¹⁵ , which is spoken as "nine point four six zero five three times ten to the fifteenth." This format is obviously much easier to both read and remember than the everyday notation.

The "9.46053" portion of the number is called the "mantissa". This part of the notation provides the necessary digits and indicates the precision of the number. In this case the number is written with six significant digits. One digit is usually placed to the left of the decimal point. The second part of the number is called the "exponent". It indicates how many times the mantissa should be multiplied by ten or, alternatively, how many places the decimal point should be moved to the right in the mantissa.

In order to understand the exponential part of the number, one needs to know and understand, first, that "10³ " means "10 x 10 x 10", both of which equal 1,000 or one thousand, and second that the pattern of exponents of ten and their meaning is as follows.

10⁴ = 10,000 (or, 1 followed by four zeros).

10³ = 1,000 (or, 1 followed by three zeros).

10² = 100 (or 1 followed by two zeros.

10¹ = 10 (or 1 followed by one zero).

10⁰ = 1 (or 1 followed by no zeros).

10^-1 = 0.1 (the 1 is placed in the first position to the right of the decimal point).

10^-2 = 0.01 (the 1 is placed in the second position to the right of the decimal point).

Etc., etc., etc..

Note also the following examples.

3.14 x 10² = 3.14 x 100 = 314.

9.46 x 10⁵ = 9.46 x 100,000 = 946,000.

8.945 x 10^-3 = 8.945 x 0.001 = 0.008945.

One of the beauties of scientific notation is that it makes calculation much easier, even with an electronic calculator. Suppose we wanted to multiply two very large numbers, for example,

836,000,000,000 x 27,930,000,000,000,000.

Even with an electronic calculator, this would be a laborious job, and there would be a very high likelihood that we would make a mistake with the number of zeros, leading to an answer that would be either ten times too big or ten times too small. With scientific notation, however, the process is much easier.

( 8.36 x 10¹¹ ) ( 2.793 x 10¹⁶ ) = 2.33 x 10²⁸ .

Doing such a calculation with a scientific calculator is a breeze, and it is even an easy job with a standard calculator. With a standard calculator, what one does is multiply the two mantissas (getting 23.35), and then add the two exponents, (getting 27), which is the new exponent. So the answer is 23.3 x 10²⁷ , which we then rewrite so that there is one digit to the left of the decimal point, and we add one to the exponent. The reason for this last is that when we changed 23.3 to 2.33, we divided by 10. In order to keep the same value for the overall number, we had to multiply the exponential part of the number by 10. This is done by simply adding one to the exponent, making the final answer 2.33 x 10²⁸ .

Dividing is almost as easy. If we wanted to divide 8.36 x 10¹¹ by 2.793 x 10¹⁶ , we would simply divide 8.36 by 2.793 (getting 2.99) and then subtract the exponents algebraically. In this case, 11 - 16 equals -5, which shouldn't surprise us because our divisor is much bigger than our dividend. This would give us:

( 8.36 x 10¹¹ ) / ( 2.793 x 10¹⁶ ) = 2.99 x 10^-5 , or 0.0000299.

So, calculating with numbers written in scientific notation is easy. All one has to do is observe the following rules.

To multiply, multiply the mantissas and add the exponents algebraically.

To divide, divide one mantissa by the other and subtract the exponents algebraically.

Also, remember that your answer should have the same number of significant digits as the number with the smallest number of significant digits that you used in the calculation.

To raise a number written in scientific notation to a power, one must raise the mantissa to that power, and then multiply the exponent of the ten by that power. For example, if one wished to raise 2.364 x 10⁴ to the fifth power, one would multiply 2.364 by itself five times (getting 73.83 and keeping four significant figures) and then multiply the exponent of the ten (4) by 5, obtaining 20. So,

(2.364 x 10⁴ )⁵ = 73.83 x 10²⁰ = 7.383 x 10²¹ .

To take the root of a number written in scientific notation, one must take the root of the mantissa and then divide the power of ten by the root desired. For example, if one wished to take the square root of the same number we started with above, we would find that

(2.364 x 10⁴ )^½ = 1.538 x 10² .

Raising numbers, including numbers written in scientific notation, to various powers on scientific calculators is easy; all one has to do is enter the number so it appears on the display, hit the " x^y " key, enter the desired power on the numerical keypad, and then hit the " = " key. This procedure also works for fractional powers like "3.5" or "4.2983" . Thus, the " x^y " key can be used to take roots also.

Most scientific calculators have a key (the "x^1/y " key) that is designed specifically for taking roots. All one has to do is enter the original number, hit the "x^1/y " key, then enter the root that is desired, and then hit the " = " key. For example, if one wished to take the cube root of a number, one would hit the "3" key for the root. Fractional roots, like "3.5" or "5.982" can also be taken.

Most scientific calculators also have special keys devoted to calculating "squares" and taking square roots.