### Scientific and Standard Notations

**Scientific and standard Notations**

When we think about writing very large or very small numbers in an easy and quicker way, scientific notation comes to our rescue.

Therefore, **scientific notation** is a way to represent very small or very large numbers. It is also easy to compare many such numbers when written in scientific notations

#### How scientific notation is written:

Scientific notation has two parts, first one is the coefficient and second part consists of base 10 and its exponent.

Coefficient is always a number between 1 and 10 . It may contain one or more than one significant digits with a decimal.

Base is always 10

Exponents is always an integer.

**Converting from standard notation to scientific notation**

*When we move towards left, exponent is positive.*

** When we move towards right, exponent is negative**.

Example1. Convert 450000 to scientific notation.

Solution: Since the coefficient must lie between 1 and 10 so decimal should be placed after 4. As decimal is moved 5 steps towards left we use positive exponent (5)

So the scientific notation is written as

Example2. Convert 0.000000215 to scientific notation.

Solution: This is very small number less than 1 so exponent would be negative.

Since coefficient must lie between 1 and 10 so we put decimal after first significant digit which is 2. For that we need to move decimal to 7 places right and whenever we move towards right while converting standard to scientific notation, we use negative exponents.

So the scientific notation is written as,

Example3. Express 0.8 in scientific notation.

Solution: Since decimal should be placed after first significant digit so decimal is moved to right by only one place.

=

Or it can also be written as

Example4. Expression 30 in scientific notation.

Solution: Coefficient is the first significant digit which is 3 here, so we need to move decimal towards left by one place.

=

So 30 can be expressed in scientific notation as

**Converting from scientific notation to standard form**

*For given negative exponents we move towards left.*

*For given positive exponents we move towards right.*

Example5. Convert to standard notation.

Solution: Since decimal is not given here so we assume 4 as 4.0 and because of negative exponent(-5) we move 5 places towards left.

=

So the standard form of given number is 0.00004

Example6. Convert to standard form.

Solution: Since exponents is given positive so we move decimal towards right by 4 places.

=

So the standard form of given number is 32700.

Example7. Express to standard form.

Solution: Since exponent is 0 that means we don’t need to move decimal. Moreover so the given number remain same without any change in it.

**Practice problems:**

Express the given numbers in scientific notation.

- 850,000
- 0.0000917
- 5102000

Express the given numbers in standard form.

**Answers:**

- 0.0428
- 31620
- 0.000012