Lesson - Scientific Notation

Scientific Notation and Standard Notation

How to change between these two notations of numbers

What is scientific notation?

Scientific notation is a special way that will allow us to write very big numbers and very small numbers in a shorter format.

How can a number be written in scientific notation?

A number is written in scientific notation when a number that is at least one but less than ten is multiplied by the number 10 (ten) is raised by a certain power. Let's use the variable "n" for number and "p" for the exponent to show how it would look for scientific notation.

 $n\times10^p$  

How can a number be written in scientific notation? (continued)

•  $4.368\ \times10^2$  is one example as to how scientific notation can be written. The first part is 4.368 as it is between 1 and 10. The second part,  $10^2$ , shows that 10 is being raised by the second power as the 2 is the exponent with the 10.

•  $7.95\ \times10^{-1}$  is another example as to how a number can be written in scientific notation. The first part 7.95 shows that it is between 1 and 10. The part  $10^{-1}$  where 10 being raised to the minus one (-1) power finishes the writing this number in scientific notation.

How would you rewrite a number in scientific notation? (Example 1)

If you wanted to rewrite the number 2849.3 into scientific notation, you would move the decimal point until the decimal follows the first nonzero digit as you would read the number from left to right.
In this case, you would move the decimal point three places to the left until the decimal follows the digit 2. Therefore, in scientific notation, the number is rewritten as  $2.8493\ \times10^3$ . 

How would you rewrite a number in scientific notation? (Example 2)

If you wanted to rewrite the number 0.000756 into scientific notation, you would move the decimal point until the decimal follows the first nonzero digit as you would read the number from left to right.
In this case, you would move the decimal point four places to the right until the decimal follows the digit 7. Therefore, in scientific notation, the number is rewritten as  $7.56\ \times10^{-4}$ . 

How do you write a number from scientific notation to standard notation? (Example 1)

When it comes to writing a number from scientific notation to standard notation, you will know in which direction to move the decimal point and how many places. For example, if you wanted to write  $8.03256\ \times10^2$  in standard notation, you would have to look at the exponent 2. The exponent will tell you as to which direction and how many places to move the decimal. Once that is done, then the number is in standard notation. In this case, the decimal will be moved two places to the right since the exponent 2 is positive. After that happens, the standard form of this number will be 803.256.

How do you write a number from scientific notation to standard/decimal notation? (Example 2)

For another example, if you wanted to write  $1.794\ \times10^{-3}$  in standard notation, you would have to look at the exponent -3. In this case, the decimal will be moved three places to the left since the exponent -3 is negative. After that happens, the standard form of this number will be 0.001794. Two zeroes were added in front of the 1 because they are needed to account for placeholders for the zeroes as if you were to multiply the actual expression shown in scientific notation.

How do you write a number from scientific notation to standard/decimal notation? (Example 3)

For another example, if you wanted to write  $6.25\ \times10^3$  in standard notation, you would have to look at the exponent 3. In this case, the decimal will be moved three places to the right since the exponent 3 is positive. After that happens, the standard form of this number will be 6250. One zero was added at the end after the 5 because it is needed to account for placeholders for any added zeroes as if you were to multiply the actual expression written in scientific notation.

A Possibly "Special" Case (From Standard Notation to Scientific Notation)

Let's say if you wanted to rewrite 8.75 in scientific notation. Based on the rules, you will have a number between 1 and 10 and multiply that by 10 raised to some power. For this situation, since the decimal is already behind the first nonzero digit, it does not need to be moved in either direction. The decimal has moved zero times. Therefore, in scientific notation, 8.75 is rewritten as  $8.75\ \times10^0$  .

A Possibly "Special" Case (From Scientific Notation to Standard Notation)

Let's say if you wanted to rewrite  $6.76\ \times10^0$  in standard notation. For this situation, the decimal does not need to be moved in either direction because the exponent is zero for the base 10. Therefore,   $6.76\ \times10^0$   is rewritten as 6.76 in standard notation.