How to write small numbers with scientific notation

As mentioed in the previous section, when describing electromagnetic radiation we are often dealing with both very large and very small numbers. Therefore scientific notation is frequently used to write these numbers.

Many of the variables for radiation equations are written in scientific notation, so you should become comfortable converting numbers to and from scientific notation as well as adding, multiplying and dividing numbers in scientific notation.

In scientific notation numbers are written in this format: a x 10b

The letter a stands for a decimal number, and the letter b stands for an exponent, or power of 10. For example, the number 5000 is written in scientific notation as 5 × 103. The number 0.03 is written as 3.0 × 10-2. 

General Rules for Converting a Number into Proper Scientific Notation

1. Find all of the significant figures in the number. Re-write those digits as a number with 1 digit in front of the decimal point and the rest of the numbers after the decimal point (number greater than or equal to 1 but less than 10.)

2. Look at the new number you have written. Count the number of places you must move the decimal point in order to get back to where the decimal point was originally located.  This will be the numerical value of your exponent.

If you have to move the decimal point to the right to get the original number, the exponent will be a positive number, if you have to move the decimal point to the left to get the original number, the exponent will be a negative number.

Example: Write 3040 in scientific notation. The decimal should be between the 3 and 0, so move the decimal point 3 places to the left, so it becomes 3.04 x 103 How to Write Small Numbers with Scientific Notation

Example: Write 0.00012 in scientific notation The decimal should be between the 1 and 2, so move the decimal point 4 places to the right, so it becomes 1.2 x 10-4 How to Write Small Numbers with Scientific Notation

Converting from Scientific Notation to “Normal”

When converting from Scientific Notation to a standard or normal notation, use the value of the exponent to determine the number of places to move the decimal point. Move the decimal place to the right if the exponent is positive and move the decimal place to the left if the exponent is negative.

Example: Write 4.01 x 105 in”normal notation” Since the exponent is positive we move the decimal point five places to the right, so it becomes 401,000 How to Write Small Numbers with Scientific Notation

For additional review on Scientific Notion see: Khan Academy: Scientific Notation Introduction and Review

Working with Exponent & Numbers in Scientific Notation

Exponents and numbers in scientific notation are frequently used in remote sensing applications.

Whether solving for the frequency of electromagnetic radiation, or using Wien's Law to determining the radiation of maximum radiation, you will find multiple variables with exponents.

Therefore it's important that to understand the rules for working with exponents, this will also make solving the equations much easier.

Multiplication

To multiply two numbers expressed in scientific notation, simply multiply the coefficient (numbers in front) and then add the exponents.

Example: Calculate (5.1 x 104) x (2.5 x 103) Multiply the coefficients, then add the exponents. In this case we end up with more that one digit in front of the decimal.

Therefore we need to move the decimal to the left one place, which adds one to the exponent How to Write Small Numbers with Scientific Notation

Division

To divide two numbers expressed in scientific notation, divide the coefficient (numbers out front) and subtract the exponents.

Example: Calculate (6.2 x 106) / (3.1 x 103) Divide the coefficients, then subtract the exponents. How to Write Small Numbers with Scientific Notation

There are also exponent rules that apply to powers and negative exponents. The table below is a quick reference guide to all of the commonly exponent rules and properties

Table of Exponents rules and properties

Rule name Rule Example
Product rules a na m = a n+m 23 ⋅ 24 = 23+4 = 128
a nb n = (a b) n 32 ⋅ 42 = (3⋅4)2 = 144
Quotient rules a n / a m = a nm 25 / 23 = 25-3 = 4
a n / b n = (a / b) n 43 / 23 = (4/2)3 = 8
Power rules (bn)m = bn⋅m (23)2 = 23⋅2 = 64
bnm = b(nm) 232 = 2(32)= 512
m√(bn) = b n/m 2√(26) = 26/2 = 8
b1/n = nb 81/3 = 3√8 = 2
Negative exponents b-n = 1 / bn 2-3 = 1/23 = 0.125

Metric Prefixes and Conversions

In addition to the use of scientific notation, you will also see a variety of metric prefixes used to describe wavelength, frequency and energy.

Table of Metric Prefixes

Prefix Symbol Multiplier Exponential
peta P 1,000,000,000,000,000 1015
tera T 1,000,000,000,000 1012
giga G 1,000,000,000 109
mega M 1,000,000 106
kilo k 1,000 103
hecto h 100 102
deca da 10 101
Base Unit 1 100
deci d 0.1 10-1
centi c 0.01 10-2
milli m 0.001 10-3
micro µ 0.000001 10-6
nano n 0.000000001 10-9
pico p 0.000000000001 10-12
femto f 0.000000000000001 10-15
atto a 0.000000000000000001 10-18

In order to perform the calculations for wavelgnth and frequency, we will need to convert frequency and wavelength units to base units of Hertz and Meters. Let's look at an example.

Example: Convert 610 nanometers to meters First look at the table above to determine the conversion, in this case we see that 1 nanometer is equal to 10-9 meters. We simply multiply 610 by 10-9 How to Write Small Numbers with Scientific Notation

For more detailed examples, watch the below video.

Activity: Lab 2

  • Lab 2: Electromagnetic Radiation

← Back

Lab 2 →

Exponents: Scientific Notation

Basic RulesNegativeEng. Not'nFractional

By using exponents, we can reformat numbers. This can be helpful, in much the same way that it's helpful (that is, it's easier) to write “twelve trillion” rather than 12,000,000,000,000., or “thirty nanometers” rather than “0.00000003 meters”.

For very large or very small numbers, it is sometimes simpler to use “scientific notation” (so called, because scientists often deal with very large and very small numbers).

The format for writing a number in scientific notation is fairly simple: (first digit of the number) followed by (the decimal point) and then (all the rest of the digits of the number), times (10 to an appropriate power).

How to Write Small Numbers with Scientific Notation

The conversion is fairly simple.

This is not a very large number, but it will work nicely for an example. To convert this to scientific notation, I first convert the “124” to “1.24”. This is not the same number as what they gave me, but (1.24)(100) = 124 is, and 100 = 102.

Then, in scientific notation, 124 is written as 1.24 × 102.

Actually, converting between “regular” notation and scientific notation is even simpler than I just showed, because all you really need to do is count decimal places.

To do the conversion for the previous example, I'd count the number of decimal places I'd moved the decimal point. Since I'd moved it two places, then I'd be dealing with a power of 2 on 10.

But should it be a positive or a negative power of 2? Since the original number (124) was bigger than the converted form (1.24), then the power should be positive.

Since the exponent on 10 is positive, I know they are looking for a LARGE number, so I'll need to move the decimal point to the right, in order to make the number LARGER. Since the exponent on 10 is “12”, I'll need to move the decimal point twelve places over.

First, I'll move the decimal point twelve places over. I make little loops when I count off the places, to keep track:

How to Write Small Numbers with Scientific Notation

Then I fill in the loops with zeroes:

How to Write Small Numbers with Scientific Notation

In other words, the number is 3,600,000,000,000, or 3.6 trillion

Idiomatic note: “Trillion” means a thousand billion – that is, a thousand thousand million – in American parlance; the British-English term for the American “billion” would be “a milliard”, so the American “trillion” (above) would be a British “thousand milliard”.

In scientific notation, the number part (as opposed to the ten-to-a-power part) will be “4.36”. So I will count how many places the decimal point has to move to get from where it is now to where it needs to be:

See also:  Comma law

How to Write Small Numbers with Scientific Notation

Then the power on 10 has to be –11: “eleven”, because that's how many places the decimal point needs to be moved, and “negative”, because I'm dealing with a SMALL number.

So, in scientific notation, the number is written as 4.36 × 10–11

Since the exponent on 10 is negative, I am looking for a small number. Since the exponent is a seven, I will be moving the decimal point seven places. Since I need to move the point to get a small number, I'll be moving it to the left.

The answer is 0.000 000 42

This is a small number, so the exponent on 10 will be negative. The first “interesting” digit in this number is the 5, so that's where the decimal point will need to go. To get from where it is to right after the 5, the decimal point will need to move nine places to the right. (Count 'em out, if you're not sure!)

Then the power on 10 will be a negative 9, and the answer is 5.78 × 10–9

This is a large number, so the exponent on 10 will be positive. The first “interesting” digit in this number is the leading 9, so that's where the decimal point will need to go. To get from where it is to right after the 9, the decimal point will need to move seven places to the left.

Then the power on 10 will be a positive 7, and the answer is 9.3 × 107

Remember: However many spaces you moved the decimal, that's the power on 10. If you have a small number in decimal form (smaller than 1, in absolute value), then the power is negative for the scientific notation; if it's a large number in decimal (bigger than 1, in absolute value), then the exponent is positive for the scientific notation.

Warning: A negative on an exponent and a negative on a number mean two very different things! For instance:

–0.00036 = –3.6 × 10–4 0.00036 = 3.6 × 10–4 36,000 = 3.6 × 104 –36,000 = –3.6 × 104

Don't confuse these!

You can use the Mathway widget below to practice converting a regular number into scientific notation. Try the entered exercise, or type in your own exercise. Then click the “paper-airplane” button to compare your answer to Mathway's. (Or skip the widget and continue with the lesson.)

Please accept “preferences” cookies in order to enable this widget.

How to Write Numbers in Scientific Notation

Scientific notation is a standard way of writing very large and very small numbers so that they’re easier to both compare and use in computations. To write in scientific notation, follow the form

How to Write Small Numbers with Scientific Notation

where N is a number between 1 and 10, but not 10 itself, and a is an integer (positive or negative number).

You move the decimal point of a number until the new form is a number from 1 up to 10 (N), and then record the exponent (a) as the number of places the decimal point was moved.

Whether the power of 10 is positive or negative depends on whether you move the decimal to the right or to the left.

Moving the decimal to the right makes the exponent negative; moving it to the left gives you a positive exponent.

To see an exponent that’s positive, write 312,000,000,000 in scientific notation:

  1. Move the decimal place to the left to create a new number from 1 up to 10.

    Where’s the decimal point in 312,000,000,000? Because it’s a whole number, the decimal point is understood to be at the end of the number: 312,000,000,000.

    So, N = 3.12.

    • Determine the exponent, which is the number of times you moved the decimal.
    • In this example, you moved the decimal 11 times; also, because you moved the decimal to the left, the exponent is positive. Therefore, a = 11, and so you get
  2. Put the number in the correct form for scientific notation

    How to Write Small Numbers with Scientific Notation

To see an exponent that’s negative, write .00000031 in scientific notation.

  1. Move the decimal place to the right to create a new number from 1 up to 10.

    So, N = 3.1.

    1. Determine the exponent, which is the number of times you moved the decimal.
    2. In this example, you moved the decimal 7 times; also, because you moved the decimal to the right, the exponent is negative. Therefore, a = –7, and so you get
  2. Put the number in the correct form for scientific notation

    How to Write Small Numbers with Scientific Notation

Scientific Notation

Scientific Notation (also called Standard Form in Britain) is a special way of writing numbers:

Like this:
Or this:

It makes it easy to use big and small values.

OK, How Does it Work?

  • Why is 700 written as 7 × 102 in Scientific Notation ?
  • 700 = 7 × 100
  • and 100 = 102 (see powers of 10)
  • so 700 = 7 × 102
  • Both 700 and 7 × 102 have the same value, just shown in different ways.

    1,000,000,000 = 109 ,
so 4,900,000,000 = 4.9 × 109 in Scientific Notation

The number is written in two parts:

  • Just the digits, with the decimal point placed after the first digit, followed by
  • × 10 to a power that puts the decimal point where it should be
    (i.e. it shows how many places to move the decimal point).

In this example, 5326.6 is written as 5.3266 × 103,
because 5326.6 = 5.3266 × 1000 = 5.3266 × 103

Try It Yourself

Enter a number and see it in Scientific Notation:

Now try to use Scientific Notation yourself:

Other Ways of Writing It

3.1 × 10^8

We can use the ^ symbol (above the 6 on a keyboard), as it is easy to type.

Example: 3 × 10^4 is the same as 3 × 104

  • 3 × 10^4 = 3 × 10 × 10 × 10 × 10 = 30,000

How to Write Small Numbers with Scientific Notation

Calculators often use “E” or “e” like this:

Example: 6E+5 is the same as 6 × 105

  • 6E+5 = 6 × 10 × 10 × 10 × 10 × 10 = 600,000

Example: 3.12E4 is the same as 3.12 × 104

  • 3.12E4 = 3.12 × 10 × 10 × 10 × 10 = 31,200

To figure out the power of 10, think “how many places do I move the decimal point?”

When the number is 10 or greater, the decimal point has to move to the left, and the power of 10 is positive.
When the number is smaller than 1, the decimal point has to move to the right, so the power of 10 is negative.

Because 0.0055 = 5.5 × 0.001 = 5.5 × 10-3

We didn't have to move the decimal point at all, so the power is 100

But it is now in Scientific Notation

Check!

After putting the number in Scientific Notation, just check that:

  • The “digits” part is between 1 and 10 (it can be 1, but never 10)
  • The “power” part shows exactly how many places to move the decimal point

Why Use It?

Because it makes it easier when dealing with very big or very small numbers, which are common in Scientific and Engineering work.

Example: it is easier to write (and read) 1.3 × 10-9 than 0.0000000013

It can also make calculations easier, as in this example:

Example: a tiny space inside a computer chip has been measured to be 0.00000256m wide, 0.00000014m long and 0.000275m high.

What is its volume?

Let's first convert the three lengths into scientific notation:

  • width: 0.000 002 56m = 2.56×10-6
  • length: 0.000 000 14m = 1.4×10-7
  • height: 0.000 275m = 2.75×10-4

Then multiply the digits together (ignoring the ×10s):

2.56 × 1.4 × 2.75 = 9.856

  1. Last, multiply the ×10s:
  2. 10-6 × 10-7 × 10-4 = 10-17 (easier than it looks, just add −6, −4 and −7 together)
  3. The result is 9.856×10-17 m3

It is used a lot in Science:

The Sun has a Mass of 1.988 × 1030 kg.

Easier than writing 1,988,000,000,000,000,000,000,000,000,000 kg
(and that number gives a false sense of many digits of accuracy.)

Use Scientific Notation
in Gravity Freeplay
How to Write Small Numbers with Scientific Notation
  • It can also save space! Here is what happens when you double on each square of a chess board:
  • Values are rounded off, so 53,6870,912 is shown as just 5×108
  • That last value, shown as 9×1018 is actually 9,223,372,036,854,775,808

Engineering Notation

Engineering Notation is like Scientific Notation, except that we only use powers of ten that are multiples of 3 (such as 103, 10-3, 1012 etc).

Examples:

  • 2,700 is written 2.7 × 103
  • 27,000 is written 27 × 103
  • 270,000 is written 270 × 103
  • 2,700,000 is written 2.7 × 106

Example: 0.00012 is written 120 × 10-6

Notice that the “digits” part can now be between 1 and 1,000 (it can be 1, but never 1,000).

The advantage is that we can replace the ×10s with Metric Numbers. So we can use standard words (such as thousand or million), prefixes (such as kilo, mega) or the symbol (k, M, etc)

Example: 19,300 meters is written 19.3 × 103 m, or 19.3 km

Example: 0.00012 seconds is written 120 × 10-6 s, or 120 microseconds

Copyright © 2020 MathsIsFun.com

Writing 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:

How to Write Small Numbers with Scientific Notation

So in a nutshell, scientific notation is composed of…

  • a number part called “c” (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.

How to Write Small Numbers with Scientific Notation

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 on 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 “c“. 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

Scientific Notation

Note: There are lots of numbers on this page that may be easier to read if the document is printed out.

A ScientificNotationWorksheet+Answers>scientific notation worksheet accompanies this lesson. Be sure to check it out!

Why Use Scientific Notation?

Scientific Notation was developed in order to easily represent numbers that are either very large or very small. Here are two examples of large and small numbers. They are expressed in decimal form instead of scientific notation to help illustrate the problem:

  • How to Write Small Numbers with Scientific Notation
  • The Andromeda Galaxy (the closest one to our Milky Way galaxy) contains at least 200,000,000,000 stars.

On the other hand, the weight of an alpha particle, which is emitted in the radioactive decay of Plutonium-239, is 0.000,000,000,000,000,000,000,000,006,645 kilograms.

As you can see, it could get tedious writing out those numbers repeatedly. So, a system was developed to help represent these numbers in a way that was easy to read and understand: Scientific Notation.

What is Scientific Notation?

Using one of the above examples, the number of stars in the Adromeda Galaxy can be written as:

2.0 x 100,000,000,000

It is that large number, 100,000,000,000 which causes the problem. But that is just a multiple of ten. In fact it is ten times itself eleven times:

10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 100,000,000,000

A more convenient way of writing 100,000,000,000 is 1011. The small number to the right of the ten is called the “exponent,” or the “power of ten.” It represents the number of zeros that follow the 1.

Though we think of zero as having no value, zeroes can make a number much bigger or smaller. Think about the difference between 10 dollars and 100 dollars.

Any one who has balanced a checkbook knows that one zero can make a big difference in the value of the number. In the same way, 0.1 (one-tenth) of the US military budget is much more than 0.01 (one-hundredth) of the budget.

(Though either one is probably more money than most of us will ever see in our checkbooks!)

  1. So we would write 200,000,000,000 in scientific notation as:
  2. 2.0 x 1011
  3. This number is read as follows: “two point zero times ten to the eleventh.”

How Does Scientific Notation Work?

  • As we said above, the exponent refers to the number of zeros that follow the 1. So:
  • 101 = 10;
    102 = 100;
    103 = 1,000, and so on.
  • Similarly, 100 = 1, since the zero exponent means that no zeros follow the 1.
  • Negative exponents indicate negative powers of 10, which are expressed as fractions with 1 in the numerator (on top) and the power of 10 in the denominator (on the bottom).
  • So:
    10-1 = 1/10;
    10-2 = 1/100;
    10-3 = 1/1,000, and so on.
  • This allows us to express other small numbers this way. For example:

2.5 x 10-3 = 2.5 x 1/1,000 = 0.0025

Every number can be expressed in Scientific Notation. In our first example, 200,000,000,000 should be written as 2.0 x 1011. In theory, it can be written as 20 x 1010, but by convention the number is usually written as 2.0 x 1011 so that the lead number is less than 10, followed by as many decimal places as necessary.

It is easy to see that all the variations above are just different ways to represent the same number:

200,000,000,000 =
20 x 1010 (20 x 10,000,000,000)
2.0 x 1011 (2.0 x 100,000,000,000)
0.2 x 1012 (.2 x 1,000,000,000,000)

This illustrates another way to think about Scientific Notation: the exponent will tell you how the decimal point moves; a positive exponent moves the decimal point to the right, and a negative one moves it to the left. So for example:

4.0 x 102 = 400 (2 places to the right of 4);

while
4.0 x 10-2 = 0.04 (2 places to the left of 4).

Note that Scientific Notation is also sometimes expressed as E (for exponent), as in 4 E 2 (meaning 4.0 x 10 raised to 2). Similarly 4 E -2 means 4 times 10 raised to -2, or = 4 x 10-2 = 0.04. This method of expression makes it easier to type in scientific notation.

2.1: Scientific Notation – Writing Large and Small Numbers

Learning Objectives

  • To express a large number or a small number in scientific notation.
  • To carry out arithmetical operations and express the final answer in scientific notation

Chemists often work with numbers that are exceedingly large or small.

For example, entering the mass in grams of a hydrogen atom into a calculator would require a display with at least 24 decimal places. A system called scientific notation avoids much of the tedium and awkwardness of manipulating numbers with large or small magnitudes.

In scientific notation, these numbers are expressed in the form

[ N imes 10^n]

where N is greater than or equal to 1 and less than 10 (1 ≤ N < 10), and n is a positive or negative integer (100 = 1). The number 10 is called the base because it is this number that is raised to the power (n). Although a base number may have values other than 10, the base number in scientific notation is always 10.

A simple way to convert numbers to scientific notation is to move the decimal point as many places to the left or right as needed to give a number from 1 to 10 (N). The magnitude of n is then determined as follows:

  • If the decimal point is moved to the left n places, n is positive.
  • If the decimal point is moved to the right n places, n is negative.

Another way to remember this is to recognize that as the number N decreases in magnitude, the exponent increases and vice versa. The application of this rule is illustrated in Example (PageIndex{1}).

Example (PageIndex{1}): Expressing Numbers in Scientific Notation

Convert each number to scientific notation.

  1. 637.8
  2. 0.0479
  3. 7.86
  4. 12,378
  5. 0.00032
  6. 61.06700
  7. 2002.080
  8. 0.01020

SOLUTION

  ExplanationAnswer
a To convert 637.8 to a number from 1 to 10, we move the decimal point two places to the left: 637.8 Because the decimal point was moved two places to the left, n = 2. (6.378 imes 10^2)
b To convert 0.0479 to a number from 1 to 10, we move the decimal point two places to the right: 0.0479 Because the decimal point was moved two places to the right, n = −2. (4.79 imes 10^{−2})
c This is usually expressed simply as 7.86. (Recall that 100 = 1.) (7.86 imes 10^0)
d Because the decimal point was moved four places to the left, n = 4. (1.2378 imes 10^4)
e Because the decimal point was moved four places to the right, n = −4. (3.2 imes 10^{−4})
f Because the decimal point was moved one place to the left, n = 1. (6.106700 imes 10^1)
g Because the decimal point was moved three places to the left, n = 3. (2.002080 imes 10^3)
h Because the decimal point was moved two places to the right, n = -2. (1.020 imes 10^{−2})

Before numbers expressed in scientific notation can be added or subtracted, they must be converted to a form in which all the exponents have the same value. The appropriate operation is then carried out on the values of N. Example (PageIndex{2}) illustrates how to do this.

Example (PageIndex{2}): Expressing Sums and Differences in Scientific Notation

Carry out the appropriate operation and then express the answer in scientific notation.

  1. ( (1.36 imes 10^2) + (4.73 imes 10^3)
    onumber)
  2. ((6.923 imes 10^{−3}) − (8.756 imes 10^{−4})
    onumber)

SOLUTION

  ExplanationAnswer
a Both exponents must have the same value, so these numbers are converted to either ((1.36 imes 10^2) + (47.3 imes 10^2) = (1.36 + 47.3) imes 10^2 = 48.66 × 10^2) or ((0.136 imes 10^3) + (4.73 imes 10^3) = (0.136 + 4.73) imes 10^3) = 4.87 imes 10^3). Choosing either alternative gives the same answer, reported to two decimal places: In converting 48.66 × 102 to scientific notation, (n) has become more positive by 1 because the value of (N) has decreased. (4.87 imes 10^3)
b Converting the exponents to the same value gives either ((6.923 imes 10^{-3}) − (0.8756 imes 10^{-3}) = (6.923 − 0.8756) imes 10^{−3}) or ((69.23 imes 10^{-4}) − (8.756 imes 10^{-4}) = (69.23 − 8.756) imes 10^{−4} = 60.474 imes 10^{−4}). In converting 60.474 × 10-4 to scientific notation, (n) has become more positive by 1 because the value of (N) has decreased. (6.047 imes 10^{−3})

When multiplying numbers expressed in scientific notation, we multiply the values of (N) and add together the values of (n).

Conversely, when dividing, we divide (N) in the dividend (the number being divided) by (N) in the divisor (the number by which we are dividing) and then subtract n in the divisor from n in the dividend.

In contrast to addition and subtraction, the exponents do not have to be the same in multiplication and division. Examples of problems involving multiplication and division are shown in Example (PageIndex{3}).

Example (PageIndex{3}): Expressing Products and Quotients in Scientific Notation

Perform the appropriate operation and express your answer in scientific notation.

  1. ([ (6.022 imes 10^{23})(6.42 imes 10^{−2})
    onumber)
  2. ( dfrac{ 1.67 imes 10^{-24} }{ 9.12 imes 10 ^{-28} }
    onumber )
  3. ( dfrac{ (6.63 imes 10^{−34})(6.0 imes 10) }{ 8.52 imes 10^{−2}}
    onumber )

SOLUTION

  ExplanationAnswer
a In multiplication, we add the exponents: [(6.022 imes 10^{23})(6.42 imes 10^{−2})= (6.022)(6.42) imes 10^{[23 + (−2)]} = 38.7 imes 10^{21}
onumber] In converting (38.7 imes 10^{21}) to scientific notation, (n) has become more positive by 1 because the value of (N) has decreased.
(3.87 imes 10^{22})
b In division, we subtract the exponents: [{1.67 imes 10^{−24} over 9.12 imes 10^{−28}} = {1.67 over 9.12} imes 10^{[−24 − (−28)]} = 0.183 imes 10^4
onumber] In converting (0.183 imes 10^4) to scientific notation, (n) has become more negative by 1 because the value of (N) has increased.
( 1.83 imes 10^3)
c This problem has both multiplication and division: [ {(6.63 imes 10^{−34})(6.0 imes 10) over (8.52 imes 10^{−2})} = {39.78 over 8.52} imes 10^{[−34 + 1 − (−2)]}
onumber ]
( 4.7 imes 10^{-31})

This page was constructed from content via the following contributor(s) and edited (topically or extensively) by the LibreTexts development team to meet platform style, presentation, and quality:

  • Marisa Alviar-Agnew (Sacramento City College)
  • Henry Agnew (UC Davis)

Scientific Notation

Learning Objectives

  • Define decimal and scientific notation
  • Convert between scientific and decimal notation
  • Multiply and divide numbers expressed in scientific notation
  • Solve application problems involving scientific notation

Before we can convert between scientific and decimal notation, we need to know the difference between the two. Scientific notation is used by scientists, mathematicians, and engineers when they are working with very large or very small numbers. Using exponential notation, large and small numbers can be written in a way that is easier to read.

When a number is written in scientific notation, the exponent tells you if the term is a large or a small number. A positive exponent indicates a large number and a negative exponent indicates a small number that is between 0 and 1. It is difficult to understand just how big a billion or a trillion is. Here is a way to help you think about it.

Word How many thousands Number Scientific Notation
million 1000 x 1000 = a thousand thousands 1,000,000  [latex]10^6[/latex]
billion (1000 x 1000) x 1000 = a thousand millions 1,000,000,000   [latex]10^9[/latex]
trillion (1000 x 1000 x 1000) x 1000 = a thousand billions  1,000,000,000,000   [latex]10^{12}[/latex]

1 billion can be written as 1,000,000,000 or represented as [latex]10^9[/latex]. How would 2 billion be represented? Since 2 billion is 2 times 1 billion, then 2 billion can be written as [latex]2 imes10^9[/latex].

A light year is the number of miles light travels in one year, about 5,880,000,000,000.  That’s a lot of zeros, and it is easy to lose count when trying to figure out the place value of the number. Using scientific notation, the distance is [latex]5.88 imes10^{12}[/latex] miles.

The exponent of 12 tells us how many places to count to the left of the decimal. Another example of how scientific notation can make numbers easier to read is the diameter of a hydrogen atom, which is about 0.00000005 mm, and in scientific notation is [latex]5 imes10^{-8}[/latex] mm.

In this case the [latex]-8[/latex] tells us how many places to count to the right of the decimal.

Outlined in the box below are some important conventions of scientific notation format.

A positive number is written in scientific notation if it is written as [latex]a imes10^{n}[/latex] where the coefficient a is [latex]1leq{a}

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