# Scientific Notation

After studying this lesson, you will be able to:

- Convert standard notation to scientific notation.
- Convert scientific notation to standard notation

**Standard Notation** is the "normal"
way of writing a number.

**Example: **45,800

**Example:** 0.034

**Scientific Notation** is the way of writing a
number as the product of a power of 10 and a number between 1 and
10. Scientific notation is often used to express very large
numbers (using positive exponents) or very small numbers (using
negative exponents.)

**Example: **4.58 Ã— 10^{ 4}

**Example:** 3.4 Ã— 10^{ -2}

**Example 1 **

Write 1.25 Ã— 10^{ 3} in standard notation.

To convert we move the decimal the same number of places as
the exponent. Since multiplying by 10^{ 3} is the same as
multiplying by 1000, we can just move the decimal three places to
the right. (Hint: if we have positive exponents, we move the
decimal to the right because we are multiplying.)

1.25 Ã— 10^{ 3} will be 1250 (we moved the decimal 3
places to the right)

**Example 2 **

Write 7 Ã— 10^{ 5} in standard notation.

To convert we move the decimal the same number of places as
the exponent. Since multiplying by 10^{ 5} is the same as
multiplying by 100,000, we can just move the decimal five places
to the right.

7 Ã— 10^{ 5} will be 700,000 (we moved the decimal 5
places to the right)

**Example 3 **

Write 4.8 x 10^{ -3} in standard notation.

To convert we move the decimal the same number of places as
the exponent. Multiplying by a negative power of 10 is the same
as dividing by that power of ten. Since multiplying by 10^{ -3}
is the same as dividing by 1000, we can just move the decimal
five places to the left. (Hint: if we have negative exponents, we
move the decimal to the left because we are dividing.)

4.8 Ã— 10^{ -3} will be .0048 (we moved the decimal 3
places to the left)

**Example 4 **

Write 1.8 Ã— 10^{ -4} in standard notation.

To convert we move the decimal the same number of places as
the exponent. Multiplying by a negative power of 10 is the same
as dividing by that power of ten. Since multiplying by 10^{ -4
}is the same as dividing by 10000, we can just move the
decimal four places to the left. (Hint: if we have negative
exponents, we move the decimal to the left because we are
dividing.)

1.8 Ã— 10^{ -4} will be .00018 (we moved the decimal 4
places to the left)

**Example 5**

Write 12,450 in scientific notation. When converting from
standard notation to scientific notation, the first thing we do
is to move the decimal in 12,450 to a point that the new number
will be between 1 and 10. That means we need to move the decimal
to where it is between the 1 and the 2. That means we moved the
decimal 4 places to the left. Now, we write the product of 1.245
and a power of 10. Our power of 10 will be 3 since we moved the
decimal 3 places . (Hint: when we are converting numbers greater
than 1 we use positive exponents. Numbers less than 1 use
negative exponents.)

12,450 will be 1.245 Ã— 10^{ 3} (we moved the decimal
3 places to the left)

**Example 6 **

Write 139,000 in scientific notation. When converting from
standard notation to scientific notation, the first thing we do
is to move the decimal in 139,000 to a point that the new number
will be between 1 and 10. That means we need to move the decimal
to where it is between the 1 and the 3. That means we moved the
decimal 5 places to the left. Now, we write the product of 1.39
and a power of 10. Our power of 10 will be 5 since we moved the
decimal 5 places .

139,000 will be 1.39 Ã— 10^{ 5} (we moved the decimal
5 places to the left)

**Example 7 **

Write 0.2362 in scientific notation. When converting from
standard notation to scientific notation, the first thing we do
is to move the decimal in 0.2362 to a point that the new number
will be between 1 and 10. That means we need to move the decimal
to where it is between the 2 and the 3. That means we moved the
decimal 1 place to the right. Now, we write the product of 2.362
and a power of 10. Our power of 10 will be -1 since we moved the
decimal 1 place . (Hint: when we are converting numbers less than
1 we use negative exponents.)

0.2362 will be 2.362 Ã— 10^{ -1} (we moved the decimal
1 place to the right) ?

**Example 8 **

Write 0.02004 in scientific notation. When converting from
standard notation to scientific notation, the first thing we do
is to move the decimal in 0.02004 to a point that the new number
will be between 1 and 10. That means we need to move the decimal
to where it is between the 2 and the 0. That means we moved the
decimal 2 places to the right. Now, we write the product of 2.004
and a power of 10. Our power of 10 will be -2 since we moved the
decimal 2 places . (Hint: when we are converting numbers less
than 1 we use negative exponents.)

0.02004 will be 2.004 Ã— 10^{ -2} (we moved the
decimal 2 places to the right)