The big and small of scientific numbers

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).

The Big and Small of Scientific Numbers

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:

The Big and Small of Scientific Numbers

Then I fill in the loops with zeroes:

The Big and Small of Scientific Numbers

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:

The Big and Small of Scientific Numbers

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.

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

The Big and Small of Scientific Numbers

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
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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
The Big and Small of Scientific Numbers
  • 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

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:

  • The Big and Small of Scientific Numbers
  • 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:

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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.

Scientific Notation in Chemistry

Learning Objectives

  • Define scientific (exponential) notation.
  • Use this notation to simplify very large or very small numbers.

The Big and Small of Scientific Numbers

Astronomers are used to really big numbers. While the moon is only 406,697 km from earth at its maximum distance, the sun is much further away (150 million km).

Proxima Centauri, the star nearest the earth, is 39,900,000,000,000 km away and we have just started on long distances. On the other end of the scale, some biologists deal with very small numbers: a typical fungus could be as small as 30 μmeters (0.000030 meters) in length and a virus might only be 0.03 μmeters (0.00000003 meters) long.

Scientific notation is a way to express numbers as the product of two numbers: a coefficient and the number 10 raised to a power. It is a very useful tool for working with numbers that are either very large or very small. As an example, the distance from Earth to the Sun is about 150,000,000,000 meters—a very large distance indeed.

In scientific notation, the distance is written as 1.5 × 1011 m. The coefficient is the 1.5 and must be a number greater than or equal to 1 and less than 10. The power of 10, or exponent, is 11 because you would have to multiply 1.5 by 1011 to get the correct number. Scientific notation is sometimes referred to as exponential notation.

A summary of SI units is given in Table below.

SI Prefixes

Prefix Unit Abbrev. Exponential Factor Meaning Example
giga G 109 1,000,000,000 1 gigameter (Gm) = 109 m
mega M 106 1,000,000 1 megameter (Mm) = 106 m
kilo k 103 1000 1 kilometer (km) = 1000 m
hecto h 102 100 1 hectometer (hm) = 100 m
deka da 101 10 1 dekameter (dam) = 10 m
100 1 1 meter (m)
deci d 10−1 1/10 1 decimeter (dm) = 0.1 m
centi c 10−2 1/100 1 centimeter (cm) = 0.01 m
milli m 10−3 1/1000 1 millimeter (mm) = 0.001 m
micro μ 10−6 1/1,000,000 1 micrometer (μm) = 10−6 m
nano n 10−9 1/1,000,000,000 1 nanometer (nm) = 10−9 m
pico p 10−12 1/1,000,000,000,000 1 picometer (pm) = 10−12 m

When working with small numbers, we use a negative exponent. So 0.1 meters is 1 × 10−1 meters, 0.01 is 1 × 10−2 and so forth. Table above gives examples of smaller units. Note the use of the leading zero

Scientific notation

Method of writing numbers, especially very large or small ones
This article is about a numeric notation. For the musical notation, see Scientific pitch notation.
“E notation” redirects here. For the series of preferred numbers, see E series.

Scientific notation (also referred to as scientific form or standard index form, or standard form in the UK) is a way of expressing numbers that are too big or too small to be conveniently written in decimal form.

It is commonly used by scientists, mathematicians and engineers, in part because it can simplify certain arithmetic operations. On scientific calculators it is usually known as “SCI” display mode.

Decimal notation

Scientific notation

2 2×100
300 3×102
4321.768 4.321768×103
−53000 −5.3×104
6720000000 6.72×109
0.2 2×10−1
987 9.87×102
0.00000000751 7.51×10−9

In scientific notation, all numbers are written in the form

m × 10n

(m times ten raised to the power of n), where the exponent n is an integer, and the coefficient m is any real number. The integer n is called the
order of magnitude and the real number m is called the significand or mantissa.

[1] However, the term “mantissa” may cause confusion because it is the name of the fractional part of the common logarithm. If the number is negative then a minus sign precedes m (as in ordinary decimal notation).

In normalized notation, the exponent is chosen so that the absolute value (modulus) of the significand m is at least 1 but less than 10.

Decimal floating point is a computer arithmetic system closely related to scientific notation.

Normalized notation

Main article: Normalized number

Any given real number can be written in the form m×10^n in many ways: for example, 350 can be written as 3.5×102 or 35×101 or 350×100.

In normalized scientific notation (called “standard form” in the UK), the exponent n is chosen so that the absolute value of m remains at least one but less than ten (1 ≤ |m| < 10). Thus 350 is written as 3.5×102.

This form allows easy comparison of numbers, as the exponent n gives the number's order of magnitude. It is the form that is required when using tables of common logarithms. In normalized notation, the exponent n is negative for a number with absolute value between 0 and 1 (e.g. 0.

5 is written as 5×10−1). The 10 and exponent are often omitted when the exponent is 0.

Normalized scientific form is the typical form of expression of large numbers in many fields, unless an unnormalized form, such as engineering notation, is desired.

Normalized scientific notation is often called exponential notation—although the latter term is more general and also applies when m is not restricted to the range 1 to 10 (as in engineering notation for instance) and to bases other than 10 (for example, 3.15×2^20).

Engineering notation

Main article: Engineering notation

Engineering notation (often named “ENG” display mode on scientific calculators) differs from normalized scientific notation in that the exponent n is restricted to multiples of 3. Consequently, the absolute value of m is in the range 1 ≤ |m| < 1000, rather than 1 ≤ |m

Big Numbers and Scientific Notation

Skip to Main ContentSkip to Navigation Quantitative Skills > Teaching Methods > Teaching Quantitative Literacy >

Big Numbers and Scientific Notation

Geologic context: radioactive decay and geologic age, geologic time, metric system Jump down to: Teaching strategies | Materials & Exercises | Student Resources

The concept of very large or very small numbers is something that is difficult for many students to comprehend. In general, students have difficulty with two things when dealing with numbers that have more zeros (either before OR after the decimal point) than they are used to. They often do not understand:

  1. that “big” and “small” are relative terms,
  2. the concept of “order of magnitude”

Scientific notation is a way to assess the order of magnitude and to visually decrease the zeros that the student sees.

It also may help students to compare very large (or very small numbers) But students still have little intuition about scientific notation.

Teaching them to recognize that scientific notation is a short hand way to better understand big and small numbers can be useful to them in all aspects of their academic career.

In science, often we work with very large or very small numbers. For example, in geology,

the age of the Earth = 4,600,000,000 years old,

or, in chemistry,

one a.m.u. = 0.00000000000000000000000000166 kilograms.

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)

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