Scientific Notation – Working With Exponents


Calculators often indicate scientific notation with capital letter E or EE.
Calculators often indicate scientific notation with capital letter E or EE. (Ray Reyes)

Scientific notation is a method of writing very large or very small numbers in decimal form. The method is used by scientists, engineers, and mathematicians. Scientific notation is also called standard form, standard index form, or scientific form. Examples of large numbers written in scientific notation include Avogadro’s number (6.022 x 1023) and the speed of light (3.0 x 108 m/s). An example of a small number written using scientific notation is the electrical charge of an electron (1.602 x 10-19 Coulombs).

Scientific Notation Basics

In scientific notation, numbers are written in two parts using the form
m × 10n
where m is any real number and n is an integer exponent. The real number is called the mantissa or significand, while the exponent is called the mantissa.

There are multiple ways to write any real number in scientific notation. For example:
250 = 2.5 x 102 = 25 × 101 = 250 × 100 = 2500 x 10-1
0.014 = 1.4 x 10-2 = 14 x 10-3 = 0.0014 x 101

Numbers are usually reported using normalized scientific notation or the standard form, where m is a decimal number greater than or equal to 1 but less than 10. Examples of numbers written in the standard form include 2.5 x 102 and 1.4 x 10-2. Even though you may report numbers this way, it’s important to know how to convert to scientific notation using other exponents so that you can perform calculations with two numbers both written in scientific notation.

To write a large number in standard form, move the decimal point to the left until only one digit remains to the left of the decimal point. Large numbers always have positive exponents. For example:

3,454,000 = 3.454 x 106

For small numbers, move the decimal point to the right until only one digit remains to the left of the decimal point. Small numbers always have negative exponents in standard form. For example:

0.0000005234 = 5.234 x 10-7

Addition and Subtraction Using Scientific Notation

Handle addition and subtraction the same way:

  • Write the two numbers in scientific notation. First make certain the two numbers have the same exponent as each other. If they do not, convert one of them to match the other. It doesn’t matter which one you convert.
  • Add or subtract the first part of the numbers, leaving the exponent portion unchanged.
  • Report the number in the standard form of scientific notation.

Examples:
(1.1 x 103) + (2.1 x 103) = 3.2 x 103
(5.3 x 10-4) – (2.2 x 10-4) = (5.3 – 1.2) x 10-4 = 3.1 x 10-4

Multiplication and Division With Scientific Notation

You don’t need the exponents to be the same for multiplication and division.

  • Multiply (or divide) the first part of the two numbers.
  • For multiplication, add the exponents. For division, subtract the exponents.
  • Convert the answer to standard form.

Multiplication Example

(2.3 x 105)(5.0 x 10-12) =

Multiply 2.3 and 5.3 to get 11.5. Add the exponents to get 10-7. At this point, the answer is:

11.5 x 10-7

You want to express your answer in standard scientific notation, which has only one digit to the left of the decimal point, so the answer should be rewritten as:

1.15 x 10-6

Division Example

(2.1 x 10-2) / (7.0 x 10-3) = 0.30 x 101 = 3.0 x 10-1

Be sure to watch your significant figures!

Using Scientific Notation on Your Calculator

A basic calculator can’t handle scientific notation. To use it, you need a scientific calculator. The method to enter numbers varies by manufacturer and model. To enter in the numbers, look for a ^ button, which means “raised to the power of” or else yx or xy, which means y raised to the power x or x raised to the y, respectively. Another common button is 10x, which makes scientific notation easy. The way these button function depends on the brand of calculator, so you’ll need to either read the instructions or else test the function. You will either press 10x and then enter your value for x or else you enter the x value and then press the 10x button. Test this with a number you know, to get the hang of it.

Calculator brands handle order of operations differently from one another. Ideally, exponent operations come before multiplication and division, which come before addition and subtraction. But, it’s a good idea to use parentheses, when available, to force the order of operations. Practice entering simple equations (or read the manual) to make certain calculations are carried out correctly.

E Notation

E notation for scientific notation
This is how Avogadro’s number looks in E notation on a calculator. (PRHaney)

Some calculators and most computers use a key labeled E, EE, EXP (for exponent), or EEX (for enter exponent) because their display cannot display superscript exponents. The capital letter E is used to distinguish scientific notation from the natural logarithm e. Usually, scientific notation is entered using the format mEn to signify m x 10n. Certain calculators, such as the popular TI-84, offer options to display values using E notation or superscripts.

References

  • Lide, David R. (ed.). (June 6, 2000). Handbook of Chemistry and Physics (81st ed.). CRC. ISBN 978-0-8493-0481-1.
  • McCalla, J.; Edwards, C.C. (2013). “How to Work in Scientific Notation on the TI-84 Plus Calculator.” TI-84 Plus Graphing Calculator for Dummies (2nd ed.). ISBN: 9781118592151
  • Mohr, Peter J.; Newell, David B.; Taylor, Barry N. (July–September 2016). “CODATA recommended values of the fundamental physical constants: 2014”. Reviews of Modern Physics. 88 (3): 035009. arXiv:1507.07956. doi:10.1103/RevModPhys.88.035009

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