# How to Write Numbers in Scientific Notation

Scientific notation allows us to express a very small or very large number in a **compact form**. The primary components of a number written in scientific notation are as follows:

So in a nutshell, scientific notation is composed of

- a
**number part**called “[latex]c[/latex]” (a number greater than or equal to 1 but less than 10)

multiplied by

- a
**number with base 10 raised to an integer power**.

The following are common numbers written in scientific notation. Try to see if you can find some pattern.

**Quick observations:**

- If a number is between 0 and 1, the exponent of base 10 is negative.

- If a number is greater than 1, the exponent of base 10 is positive.

Now let’s talk about the general steps involved in how to convert a decimal number into scientific notation.

## Steps in Writing Decimal Numbers into Scientific Notation

**STEP 1**: Identify the initial location of the original decimal point.

**STEP 2**: Identify the **final location** or “destination” of the original decimal point.

- The final location of the original decimal point must be directly
**to the right of the first nonzero number**.

**STEP 3**: Move the original decimal point to its final location.

- You will get a number here called “[latex]c[/latex]”. Its value
**must**be greater than or equal to 1, but less than 10. - When the decimal is moved
*towards the left*, the count for the exponent of base 10 should be**positive**. - When the decimal is moved
*towards the right*, the count for the exponent of base 10 should be**negative**.

**STEP 4**: Write “[latex]c[/latex]” multiplied by some power of base 10. It should look something like this: [latex]c\, \times \,{10^n}[/latex]

### Examples of How to Write Decimal Numbers into Scientific Notation

**Example 1**: Rewrite the given decimal number **5,800** in scientific notation.

We start by identifying where the original location of the decimal point, and its new location.

Now, we move the decimal point from the starting point to its final destination while counting the number of decimal places.

- Remember the rule above, if the decimal is moved
*towards the left*, the count for the exponent of base 10 is**positive**.

That makes our value of “[latex]c[/latex]” as [latex]c = 5.8[/latex], and the power of 10 is **10 ^{3}**. Putting them together in the required format, our final answer is

Always remember to make sure that “[latex]c[/latex]” value always has the **decimal point right after the first digit** which is the case here. Great!

**Example 2**: Rewrite the given decimal number **1,730,000** in scientific notation.

Begin by locating the initial decimal point, and where it is going.

It appears that we are going to move the decimal to the left. Remember, such type of movement will incur a **positive exponent** for the base 10.

The coefficient or “[latex]c[/latex]” value is [latex]c = 1.73[/latex] and the power of 10 is **10 ^{6}**. This should give us the final answer of

**Example 3**: Rewrite the given decimal number **33,335,000,000,000** in scientific notation.

The starting decimal point is on the far right. We need to move it to the **left** until we have a decimal number between 1 and 10.

Moving the decimal from right to left implies that the power of 10 will have a positive integer.

The value of the coefficient is [latex]c = 3.3335[/latex], and the power of 10 becomes **10 ^{13}**. Therefore, the final answer of our scientific notation is just

**Example 4**: Rewrite the given decimal number **0.0009** in scientific notation.

It is obvious that the original decimal point is to the left of the nonzero digit. We will move the decimal going to the **right**. The rule above states that

- When the decimal is moved
*towards the right*, the count for the exponent of base 10 should be**negative**.

Moving the decimal point to the right should yield a **negative exponent for the base 10**.

The value for [latex]c[/latex] is just [latex]c = 9[/latex], and the power of 10 is **10 ^{-4}**. Our final scientific notation answer should be

**Example 5**: Rewrite the given decimal number **0.00000000086** in scientific notation.

Maybe you can predict that since the given decimal number is between 0 and 1, we should have a scientific notation with a negative power. First, identify the initial decimal point and where it is going.

Let’s go ahead and move the original decimal point towards the right until the decimal point is to the **right of the first nonzero digit** which is 8. Going to the right means we are going to accumulate negative counts for the power of 10.

This gives us the coefficient value of [latex]c = 8.6[/latex], and the power of 10 value is **10 ^{-10}**. Writing the final scientific notation, we have

**Example 6**: Rewrite the given decimal number **0.000000000001234** in scientific notation.

The given decimal number is less than 1, so we expect to move the decimal point **towards the right** such that it *stops after the first nonzero digit*.

Let’s move the decimal point to the right, and it should accumulate a negative power of 10.

We have a coefficient value of [latex]c = 1.234[/latex], and base ten value of **10 ^{-12}**. This gives us a scientific notation of

That’s it, folks! I hope you learn the basics of how to write a decimal number into its scientific notation form.