Using Scientific Notation

What is Scientific Notation?

Scientific notation is a compact way of writing numbers that are too large or too small to be conveniently written in decimal form. It is a shorthand letting us write numbers using powers of 10. Scientific fields such as astronomy, physics, chemistry, and statistics frequently use scientific notation.

Below is an example of shorthand notation:

• 3.2 X 10⁸
• 3.2 X 10^8
• 3.2E8

All three forms of scientific notation are equivalent. In the last format, the E stands for exponent.

In this blog post, you’ll learn how to interpret scientific notation, convert numbers to this format, and how to use it for multiplication and division.

How to Do Scientific Notation

Scientific notation in math and statistics expresses enormous and miniscule numbers in a compact and easy-to-understand format. It contains the following two parts:

• The coefficient: A number between 1 and 10.
• A power of 10: Defined by an exponent.

To convert scientific notation to its numeric form, simply multiply the coefficient by the power of 10. However, you don’t actually need to perform the multiplication. Because the exponent uses a base of 10, you just move the decimal place in the coefficient to the right or left by the value of the exponent.

The exponent value indicates whether the number is very large or small by specifying the direction you need to move the decimal place:

• Positive: Move the decimal point to the right for very large numbers.
• Negative: Move the decimal point to the left for very small numbers.

Large Number Example

So, let’s convert our example scientific notation to its decimal form.

3.2 X 10⁸

The exponent is +8. Being positive, it tells us that we’re dealing with a large number and that we need to shift the decimal place to the right.

To convert this scientific notation, take our coefficient of 3.2 and then move its decimal point to the right by eight places.

3.2 X 10⁸ = 320,000,000

Try converting scientific notation to the numeric form yourself. A light year is approximately 9.46×10¹² km or 5.88×10¹² mi. How many KM or MI are in a lightyear?

Small Number Example

The diameter of a hydrogen atom is approximately 5.29 X 10^-11 meters. You might also see it expressed using 5.29E-11.

The exponent is -11. Its negative value indicates we are working with a tiny number and that we’ll need to shift the decimal place to the left.

To convert this scientific notation to its numeric form, take the coefficient of 5.29 and move the decimal point left by 11 places.

5.29 X 10^-11 = 0.0000000000529

The diameter of a hydrogen atom is approximately 0.0000000000529 meters.

Statistical software and Excel frequently report small p-values using scientific notation. For example, see Using Excel to Perform T-tests and look in the statistical output.

Converting Numbers to Scientific Notation

To convert a number from numeric form to scientific notation, follow these steps:

1. Move the decimal point so there is only one non-zero digit to the left of the decimal point. You should be able to write it as a number between 1 and 10.
2. Count the number of places you moved the decimal point.
3. Write the exponent as the number of places you moved the decimal point, with a positive exponent for numbers greater than 1 and a negative exponent for numbers less than 1.

For example, the distance from the sun to Pluto is 5900000000km on average.

To get from the numeric value to the coefficient of 5.9, we must move the decimal nine places to the left.

Because we’re dealing with a large number, we need a positive exponent. Therefore, the scientific notation is the following:

5900000000 = 5.9 X 10⁹ km

Let’s convert a small number to scientific notation. The mass of an electron is approximately 0.0000000000000000000000000000009109 kilograms.

Our coefficient will be 9.109. We need to move the decimal place to the right by 31 places to get from the numeric value to the coefficient.

Because we’re dealing with a small number, the coefficient must be negative. Consequently, the scientific notation is the following:

0.0000000000000000000000000000009109 = 9.109 X 10^-31 kg.

Multiplying Scientific Notation

Scientific notation simplifies multiplication of huge and tiny numbers. Follow these steps for multiplication:

1. Multiply the coefficients.
2. Add the exponents for the power of 10.

Suppose we are multiplying 3.2 X 10⁸ and 5.9 X 10⁹.

Start by multiplying the coefficients: 3.2 X 5.9 = 18.88.

Then add the exponents of 8 and 9 = 17.

Finally, combine the coefficient product with the power of 10:

(3.2 X 10⁸) X (5.9 X 10⁹) = 18.88 X 10¹⁷.

At this point, you can shift the decimal one more place to the left to produce a proper coefficient for scientific notation that is between 1 and 10. Be sure to increase the exponent in the scientific notation by 1 to reflect this additional shift.

18.88 X 10¹⁷ = 1.888 X 10¹⁸

Imagine we have two miniscule numbers in scientific notation: 9.109 X 10^-31 and 5.29 x 10^-11.

Multiply the coefficients: 9.109 X 5.29 = 48.18661

Add the exponents: -31 + (-11) = -42

Therefore:

(9.109 X 10^-31) X (5.29 x 10^-11) = 48.18661 X 10^-42

Like the previous example, shift the decimal one more place to the left to produce a proper coefficient and increase the exponent by 1.

48.18661 X 10^-42 = 4.818661 X 10^-41

Dividing Scientific Notation

The process for how to divide scientific notation is similar to the multiplication procedure. Follow these steps:

1. Divide the coefficients.
2. Subtract the exponents (numerator – denominator).

Suppose we want to calculate the proportion of Earth’s mass to that of the entire solar system. We need to divide the two masses.

The mass of the Earth is approximately 5.97 x 10²⁴ kilograms, and the solar system’s total mass is about 1.0014 x 10³⁰ kilograms. To find the proportion, we need to divide the following scientific notation:

(5.97 x 10²⁴ kg) / (1.0014 x 10^³⁰ kg)

To divide these numbers in scientific notation, follow these steps:

Divide the coefficients:

5.97 / 1.0014 = 5.96

Subtract the exponents:

24 – 30 = -6

Combine the coefficient and the power of 10:

5.96 x 10^-6

So, the proportion of the solar system’s mass taken up by the Earth is approximately 5.96 X 10^-6. We divided two vast numbers to obtain a tiny proportion!