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We saw in the rounding numbers section that it often makes sense to write long or complicated decimal values in condensed forms. In a similar vein, it would be nice to have a neater way to write really big or really small numbers without listing too many zeros in a row, but also without sacrificing too much information.
This is where scientific notation comes in.
Writing numbers in standard form
Scientific notation is a useful way to write large or small numbers in a compact form. It uses powers of 10 to "condense" a lot of digits. Numbers written in scientific notation are of the form
where 1≤a<10 and k∈Z.
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Scientific notation is sometimes called "standard form."
k tells you how many spaces to move the decimal point so that you get a number between 1 and 10.
Moving the decimal to the left gives a positive value for k, as in the animation above.
Moving the decimal to the right gives a negative value for k:
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Discussion
Here's an example: The distance from the Earth to the sun is about 384,400,000 meters. Using scientific notation will allow us to write this number with fewer zeros.
Think about how many digits this number has. How many digits do we need to "condense" in order to write 384,400,000 in the form a×10k?
Can you rewrite 384,400,000 using scientific notation? Why does this work?
To write 384400000 in the form a×10k with 1≤a<10, we follow two steps:
Place the decimal point just after the first non-zero digit.
Since 384400000 has nine digits, inserting a decimal after the “3” gives
which indeed satisfies 1≤a<10.
Record how many places we moved the decimal.
Originally the decimal was at the end of the integer; we moved it 8 places to the left to get 3.844. Hence
Putting these together,
Why does this work? By definition,
and multiplying 3.844 by 108 shifts its decimal point eight places to the right:
Discussion
The same idea applies to really small numbers. In this case, we can use negative exponents to condense a large number of zeros after the decimal point.
For example, the radius of a hydrogen atom is approximately 0.000000000529 meters. This is quite an unwieldy number of zeros!
Count the number of digits in 0.000000000529. Can you use your count of the digits to write 0.000000000529 in the form a×10k, where 1≤a<10?
Why does this work?
There are twelve digits after the decimal point in
Of these twelve digits, the first nine are zeros before the “529.”
To write this in the form a×10k with 1≤a<10, we choose a=5.29. We observe that
since each multiplication by 10 shifts the decimal point one place to the right, and doing so ten times moves it past the nine zeros and lands just after the 5. Equivalently,
Why does this work? Because powers of ten simply move the decimal point:
Thus expressing a very small number as a×10k keeps the same value but replaces a long string of zeros by a compact exponent.
Converting numbers into scientific notation is nice for saving space, but in order to really make it useful, we need to be able to add, subtract, multiply, and divide numbers that are given in scientific notation originally. Let's use the numbers 1.5×106 and 4×105 as an example.
In order to add these numbers, their exponents need to be the same power. We can rewrite the number with the smaller power in terms of the greater one by moving its decimal point:
Rewriting in terms of the larger power and then performing arithmetic on the coefficients also works to subtract numbers in scientific notation, like in the case of the difference 1.5×106−4×105:
If this doesn't make sense to you, it can help to think about "lining up" zeros, since the amount of leading or trailing zeros a number has determines how we write it in scientific form. The same idea applies for multiplying and dividing numbers written in scientific notation.
Multiplying & dividing numbers in scientific notation
We can multiply and divide numbers in scientific form as follows:
Multiplying and dividing numbers in scientific notation relies heavily on exponent rules.
Exercise
Let x=3.45×107,y=2.10×105.
Find the value of xy.
Find the value of yx.
Select the correct option
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