Scientific Notation (write in standard form)

Scientists often have to deal with very big numbers (how far is it from Earth to the Andromeda Galaxy?) or very tiny numbers (how big is an atom?).

As we know, big numbers have lots of zeros at the end (it's 2,538,000 light years from Earth to the Andromeda Galaxy), and small numbers require lots of zeros after the decimal point (some scientists estimate that atoms are about .00000001 centimeters in diameter.)

These zeros can make numbers really long and making a mistake with zeros (adding one or leaving one out) is a big deal.  Think about it.  If you won \$90 but accidentally forgot a zero, you'd only get \$9. If you accidentally added a zero you'd get \$900 -- those are huge differences!  Basically, every time you add or subtract a zero, it changes the number by a factor of 10. And, accuracy is critical in science.  So, scientists and mathematicians developed scientific notation, which is a way to maintain all of the details of a number, but compress the zeros.

Scientific notation uses the fact that every 0 in a number represents a factor of 10, and simply uses 10s with exponents to account for those numbers.

Earth is 2,538,000 light years from the Andromeda Galaxy.  

Scientists would write that as $\large{2.538\times 10^6}$

What does that mean?

$$\eqalign{&2.538 \times 10^6&&\text{Original scientific notation}\\=&2.538 \times 1000000&&\text{Rewrite with expanded exponent (remember} 10^6\text{ is a 1 with 6 zeros behind it)}\\=&2,538,000&&\text{Multiply decimal times expanded number}}$$

To represent tiny numbers, scientific notation uses 10s with negative exponents, which represent fractions or decimals of whole numbers.  For example $10^{-1} = \dfrac{1}{10} \text {  OR  } .01 \text{  and  } 10^{-3} = \dfrac{1}{1000} \text{  OR  } .001$.

The diameter of an atom is about .00000001 centimeters in diameter.

Scientists would write that as $\large{1.0\times 10^{-8}}$

What does that mean?

$$\eqalign{&1 \times 10^{-8}&&\text{Original scientific notation}\\=&1 \times \dfrac{1}{100000000}\text{  OR  } .00000001&&\text{Rewrite exponent as fraction or decimal}\\=&.00000001&&\text{Multiply decimal times decimal or fraction}}$$

As you can see, scientific notation does not change numbers at all, it just tries to write the zeros in a more compact form.  Numbers that are tiny have negative exponents (to show that the number should have that many zeros between it and the decimal point).  Numbers that are huge have positive exponents, showing that they should have that many zeros added to the end.

So, how can you convert numbers into and out of scientific notation?  If you get stuck, you can always do the math.  If you find a number in scientific notation, you can actually multiply the number by the exponent and get the correct answer.   But there's an easier and faster way.

Example: $\text{What is } 4.35 \times 10^5 \text{ in standard form?}$

$$\eqalign{&4.35 \times 10^5&&\text{Original scientific notation}\\=&4.35 \times 100000&&\text{Move the decimal point from its current location 5 places to the right.}\\=&4.\underrightarrow{35000.}&&\text{Take the exponent (5) and move the decimal that many places to the right, to get number in standard form.}}$$

As you can guess, you can use a similar shortcut with small numbers, but because the number is tiny and not huge, you move the decimal to the left rather than the right (move it the same number of places as the number in the exponent).

Example: $\text{What is } 7.1 \times 10^{-9} \text{ in standard form?}$

$$\eqalign{&7.1 \times 10^{-9}&&\text{Original scientific notation}\\=&7.1 \times .000000001&&\text{Move the decimal point from its current location 9 places to the left.}\\=&\underleftarrow{.000000007.}1&&\text{Take the exponent (9) and move the decimal that many places to the left, to get number in standard form.}}$$

Overall, when you are given a number in scientific notation know that it's actually either bigger or smaller than it looks.  A number with a positive exponent on the 10 is actually a larger number.  A number with a negative exponent on the 10 is actually a smaller number.

The simple rule to follow when converting these numbers to standard form.

1. Locate the decimal (in scientific notation, the decimal will always be after the first digit).
2. Locate the exponent on the 10.
   1. If it's a positive exponent, move the decimal that many places to the right (if the exponent is 5, move 5 places to the right). Remember: a positive exponent on the 10 means that the number will get bigger!
   2. If it's a negative exponent, move the decimal that many places to the left (if the exponent is 7, move 7 places to the left). Remember: a negative exponent on the 10 means that the number will get smaller!
3. Add zeros as necessary to fill in any empty places.