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:

in c × 10^n, c is any number from 1 to 10 but not including 10. on the other hand, 10 is always the base of 10 while the exponent n must always be an integer.

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.

a table showing how numbers are written in different formats such as in decimal form, as a power of 10, and in scientific notation. for example, one millionth is written in decimal form as 0.000001, 10^-6 when written as a power of 10 then in scientific notation as 1×10^-6. on the other hand, one thousand is written as 1,000 in decimal form, 10^3 as a power of 10, and written in scientific notation as 1×10^3.

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.

for the number 5,800, the original decimal point is located on the rightmost side of the number while the new location of the decimal point will be between the numbers 5 and 8.

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.

from the original location of the decimal in the number 5800, we will count and move 3 decimal places to the left to get a decimal between 1 and 10, which will give us 5.800.

That makes our value of “[latex]c[/latex]” as [latex]c = 5.8[/latex], and the power of 10 is 10³. Putting them together in the required format, our final answer is

we write the number 5,800 in scientific notation as 5.8 × 10^3. note that our exponent is positive 3 because we moved 3 decimal spaces to the left.

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.

in our decimal number 1,730,000, the starting decimal point is located on the right of the last 0 which is the rightmost side of the number. when moved to get a decimal number between 1 and 10, the final location of the decimal point is between 1 and 7.

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.

from its original location at the end of the number 1730000, we will move the decimal 6 places to the left which will give us 1.730000. as we move the decimal or dot to get a decimal that is between 1 and 10, we need to pay attention and count the number of decimal places we moved the decimal from.

The coefficient or “[latex]c[/latex]” value is [latex]c = 1.73[/latex] and the power of 10 is 10⁶. This should give us the final answer of

the decimal number 1730000, when written in scientific notation, is 1.73 × 10^6.

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.

the original location of the decimal point in the number 33,335,000,000,000 is after the last zero. to get a "c" or number between 1 and 10, we need to move the decimal point to the left ending between the first and second 3.

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

we will move the decimal point 13 decimal places to the left to get a decimal number that is between 1 and 10. remember to count the number of decimal places we moved the decimal point from, which in this case is 13 places, because we need this when writing our scientific notation.

The value of the coefficient is [latex]c = 3.3335[/latex], and the power of 10 becomes 10¹³. Therefore, the final answer of our scientific notation is just

the scientific form of the number, 33,335,000,000,000 is 3.3335 × 10^13.

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.

in the decimal number 0.0009, the original decimal point is located between the first and second zeros. to get a number between 1 and 10, we need to move the decimal point to the right which in this case, will be located to the right side of 9 or the rightmost side of our number. from 0.0009, it will become 00009.

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

we will move the decimal point 4 decimal places to the right to get a "c" or number between 1 and 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

we write the decimal number 0.0009 in scientific notation as 9 × 10^-4.

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.

in our decimal number 0.00000000086, the location of the original decimal is after the first zero. since we want to get a number from 1 to 10, we need to move the decimal point to the right ending next to the number 8. this will be the new location of our decimal point.

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.

moving 10 decimal places to the right, we'll have 00000000008.6. to continue writing in scientific notation, we will drop the zeros and leave the nonzero numbers.

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

the scientific notation form of the decimal number 0.00000000086 is 8.6 × 10^-10.

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.

we first determine the location of our original decimal point. in our given decimal number, 0.000000000001234, the original decimal is located after the first 0. we will then move the decimal point to the right to obtain a number from 1 to 10. thus, we'll have 0000000000001.234 where the new location of our decimal number is next to 1.

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

to get a decimal number between 1 and 10, we will move the decimal point 12 decimal places to the right from its original location.

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

our decimal number, 0.000000000001234, can be written in scientific notation as 1.234 × 10^-12. note that we have an exponent of -12 for our base 10. that is because we moved our decimal point 12 decimal places to the right which means our exponent will be negative.

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