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# Scientific Notation Rules for Physics Help

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By McGraw-Hill Professional
Updated on Sep 17, 2011

## Introduction

In printed literature, power-of-10 notation generally is used only when the power of 10 is large or small. If the exponent is between −2 and 2 inclusive, numbers are written out in plain decimal form as a rule. If the exponent is −3 or 3, numbers are sometimes written out and are sometimes written in power-of-10 notation. If the exponent is −4 or smaller, or if it is 4 or larger, values are expressed in power-of-10 notation as a rule.

Some calculators, when set for power-of-10 notation, display all numbers that way. This can be confusing, especially when the power of 10 is zero and the calculator is set to display a lot of digits. Most people understand the expression 8.407 more easily than 8.407000000E + 00, for example, even though they represent the same number.

With this in mind, let’s see how power-of-10 notation works when we want to do simple arithmetic using extreme numbers.

Addition of numbers is best done by writing numbers out in ordinary decimal form if at all possible. Thus, for example,

(3.045 × 10 5 ) + (6.853 × 10 6 ) = 304,500 + 6,853,000

= 7,157,500

= 7.1575 × 10 6

(3.045 × 10 −4 ) + (6.853 × 10 −7 ) = 0.0003045 + 0.0000006853

= 0.0003051853

= 3.051853 × 10 −4

(3.045 × 10 5 ) + (6.853 × 10 −7 ) = 304,500 + 0.0000006853

= 304,500.0000006853

= 3.045000000006853 × 10 5

## Subtraction

Subtraction follows the same basic rules as addition:

(3.045 × 10 5 ) − (6.853 × 10 6 ) = 304,500 − 6,853,000

= −6,548,500

= −6.548500 × 10 6

(3.045 × 10 −4 ) − (6.853 × 10 −7 ) = 0.0003045 − 0.0000006853

= 0.0003038147

= 3.038147 × 10 −4

(3.045 × 10 5 ) − (6.853 × 10 −7 ) = 304,500 − 0.0000006853

= 304,499.9999993147

= 3.044999999993147 × 10 5

Power-of-10 notation may, at first, seem to do more harm than good when it comes to addition and subtraction. However, there is another consideration: the matter of significant figures . These make addition and subtraction, in the inexact world of experimental physics, quite easy and sometimes trivial. If the absolute values of two numbers differ by very many orders of magnitude, the one with the smaller absolute value (that is, the one closer to zero) can vanish into insignificance and, for practical purposes, can be ignored. We’ll look at this phenomenon later in this chapter.

## Multiplication

When numbers are multiplied in power-of-10 notation, the decimal numbers (to the left of the multiplication symbol) are multiplied by each other. Then the powers of 10 are added. Finally, the product is reduced to standard form. Here are three examples, using the same three number pairs as before:

(3.045 × 10 5 ) × (6.853 × 10 6 ) = (3.045 × 6.853) × (10 5 × 10 6 )

= 20.867385 × 10 (5+6)

= 20.867385 × 10 11

= 2.0867385 × 10 12

(3.045 × 10 −4 ) × (6.853 × 10 −7 ) = (3.045 × 6.853) × (10 −4 × 10 −7 )

= 20.867385 × 10 [−4+(−7)]

= 20.867385 × 10 −11

= 2.0867385 × 10 −10

(3.045 × 10 5 ) × (6.853 × 10 −7 ) = (3.045 × 6.853) × (10 5 × 10 −7 )

= 20.867385 × 10 (5−7)

= 20.867385 × 10 −2

= 2.0867385 × 10 −1

= 0.20867385

This last number is written out in plain decimal form because the exponent is between −2 and 2 inclusive.

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