# What is scientific notation?

How were arithmetic problems solved before the advent of pocket calculators? In those days, there were only three methods for solving a multiplication problem that a person couldn't do in his or her head: pencil and paper, getting time on a mainframe computer, or using a slide rule. Many students would rely on a slide rule.

How does a slide rule work? First, it cannot multiply any numbers outside of a range between 1 and 9.99. So in order to multiply 187 by 26 the numbers would be rewritten so that they were something between 1 and 9.99:

Then the slide rule could be used to find:

However, this still left more work to be done to get to the final answer. Enough with the slide rule! Thank goodness for calculators. By using the slide rule, however, individuals could learn a valuable lesson about how to work with numbers of different magnitudes. This led to an appreciation for something called scientific notation.

Scientific notation is defined as a standardized way to represent any number as the product of a real number and a power of 10.

In this form, a is called the coefficient and b is the exponent.

The coefficient is defined as an integer (except zero) to the left of a decimal point plus a mantissa. A manitssa is defined as the significant digits to the right of the decimal point. Here's an example of scientific notation with the important parts of the expression labeled:

## Why Scientific Notation?

Consider the following expression, which is very important to engineers and physicists:

Look at all of those zeros. This expression is the charge on a single electron. The 'C' stands for Coulomb, which is a unit of electrical charge. In this form, it is a very cumbersome number to work with. It might be easier if it was expressed as follows:

Here's another example. Sometimes the mass of the earth, in kilograms, is used to solve gravity problems. Here it is in its raw form:

Again, this is a pretty unwieldy number. It would be nicer represented as follows:

So it can be more convenient to use scientific notation. But, what are the rules for expressing a number in scientific notation, and how are the expressions found?

### A Few Important Rules

The great thing about scientific notation is that there is a universally accepted, or standardized, form. Here are the rules:

• Most scientific calculators will put any number into scientific notation, but when it is done by hand, there is a simple procedure:
• With the number in its original form, start at the decimal point and count the spaces between digits, either left or right, until you get to the first non-zero digits.
• 45,000,000 —> 4.5000000

Follow the example using the number 45,000,000. In this case, move the decimal point until it is after the first non-zero number. If moving to the left, go one additional space so that there is a whole number on the left and a mantissa on the right.

107

The number of spaces the decimal point was moved becomes the exponent. If it was moved right, the exponent is negative. If left, the exponent is positive. If the exponent is zero, scientific notation isn't needed.

45,000,000 = 4.5 x 107

If the number (or a result) is not quite in standard form that is because the mantissa is 10 or greater or less than 1; it can be corrected. Count left or right from the decimal point until there is just one whole number on the left. If it was moved left, then add that number to the existing exponent. If right, then subtract.

## scientific-notation

Definitions

scientific notation a mathematical expression used to represent any decimal number as a number between one and ten raised to a specific power of ten (Ex.: 4.1 × 10 for 4.1, 4.1 × 10 for 41, 4.1 × 10 for 410, 4.1 × 10 for 0.41, 4.1 × 10 for 0.041): often used for approximate computations with very large or small numbers

noun

A method of writing or displaying numbers in terms of a decimal number between 1 and 10 multiplied by a power of 10. The scientific notation of 10,492, for example, is 1.0492 × 104.
A method of expressing numbers in terms of a decimal number between 1 and 10 multiplied by a power of 10. The scientific notation for 10,492, for example, is 1.0492 × 104. Noun

(uncountable)

1. (mathematics) a method of writing, or of displaying real numbers as a decimal number between 1 and 10 followed by an integer power of 10
2. (mathematics) an alternative format of such a decimal number immediately followed by E and an integerThe number 0.00236 is written in scientific notation as 2.36Ã—10âˆ’3 or as 2.36E-3.

The display of numbers in floating point form. The number (mantissa) is always equal to or greater than one and less than 10, and the base is 10. For example, 2.345E6 is equivalent to 2,345,000. The number following E (exponent) represents the power to which the base should be raised (number of zeros following the decimal point).

## 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

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
• 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

## Using Scientific Notation

Cite

Sometimes, especially when you are using a calculator, you may come up with a very long number. It might be a big number, like 2,890,000,000. Or it might be a small number, like 0.0000073.

Scientific notation is a way to make these numbers easier to work with. In scientific notation, you move the decimal place until you have a number between 1 and 10. Then you add a power of ten that tells how many places you moved the decimal.

In scientific notation, 2,890,000,000 becomes 2.89 x 109. How?

• Remember that any whole number can be written with a decimal point. For example: 2,890,000,000 = 2,890,000,000.0
• Now, move the decimal place until you have a number between 1 and 10. If you keep moving the decimal point to the left in 2,890,000,000 you will get 2.89.
• Next, count how many places you moved the decimal point. You had to move it 9 places to the left to change 2,890,000,000 to 2.89. You can show that you moved it 9 places to the left by noting that the number should be multiplied by 109.

2.89 x 109 = 2.89 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 102.89 x 109 = 2,890,000,000

Scientific notation can be used to turn 0.0000073 into 7.3 x 10-6.

• First, move the decimal place until you have a number between 1 and 10. If you keep moving the decimal point to the right in 0.0000073 you will get 7.3.
• Next, count how many places you moved the decimal point. You had to move it 6 places to the right to change 0.0000073 to 7.3. You can show that you moved it 6 places to the right by noting that the number should be multiplied by 10-6.

7.3 x 10-6 = 0.0000073

Remember: in a power of ten, the exponent—the small number above and to the right of the 10—tells which way you moved the decimal point.

• A power of ten with a positive exponent, such as 105, means the decimal was moved to the left.
• A power of ten with a negative exponent, such as 10-5, means the decimal was moved to the right.
 Powers of Ten billions109 = 1,000,000,00010 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 1,000,000,000 millions106 = 1,000,00010 x 10 x 10 x 10 x 10 x 10 = 1,000,000 hundred thousands105 = 100,00010 x 10 x 10 x 10 x 10 = 100,000 ten thousands104 = 10,00010 x 10 x 10 x 10 = 10,000 thousands103 = 1,00010 x 10 x 10 = 1,000 hundreds102 = 10010 x 10 = 100 tens101 = 10 ones100 = 1 tenths10–1 = 1/101/10 = 0.1 hundredths

## General Astronomy/Scientific Notation

In previous sections, we discussed some numbers that were very large. In astronomy, the appearance of such huge numbers is common.

This is one reason astronomers and other scientists use scientific notation when working with very large or very small numbers.

Scientific notation is a system for writing and working with numbers that makes it much easier to deal with numbers that are very small or very large.

For example, the Milky Way Galaxy contains roughly three duodecillion tons of material. That is a rather cumbersome number. (Astronomers would never actually write this. Instead, they would say that the Milky Way contains one trillion times the mass of the Sun, which is somewhat easier. We'll use this much larger number for our demonstration.) You could also write this number as

3 000 000 000 000 000 000 000 000 000 000 000 000 000 tons,

but that's even worse. Scientific notation makes the number much more compact and readable:

3 × 1039 tons.

This number is verbally expressed as “three times ten to the 39 power tons.” This is numerically equivalent to the first two expressions.

Using an exponent to represent a number allows that number to grow very big very fast. This makes it convenient to represent very large or very small numbers using exponential notation.

A number written correctly in scientific notation has two parts. The first is a number greater than or equal to 1 and less than 10 (but it can be either positive or negative).

This is sometimes called the mantissa. The second part is the number ten raised to a whole number power. The exponent of the second number is called the power.

Some examples of numbers written correctly in scientific notation are:

2 × 1018
-1.4 × 102
7.656 × 10-4
2.1 × 100

These, on the other hand, are not valid examples of numbers written in scientific notation:

0.1 × 104 is wrong because the mantissa is less than 1
12 × 103 is wrong because the mantissa is not less than 10
8.4 × 102.2 is wrong because the power is not a whole number

Remember that

10n = 10 × 10 × 10 × … for n times,

which means that ten raised to n power is the same as 10 multiplied by itself n times, which is the same as a 1 with n zeros written after it. For example, 103 is 10 × 10 × 10, or 1000. That means that our earlier number, 3 × 1039 tons, is equivalent to

3 000 000 000 000 000 000 000 000 000 000 000 000 000 tons,

which is a three followed by 39 zeros. A number written in scientific notation with a negative power corresponds to a small number. For example, the number 1 × 10−3 is written as 0.001 in conventional notation. In general,

10-n = 1/10 × 1/10 × 1/10 × … for n times.

 These are the steps in converting a large number expressed in scientific notation into standard notation. Write the mantissa and find the location of the decimal point, and shift the decimal point by the number indicated in the power. Here, the power is 9, so the decimal is moved nine digits to the right. These are the steps in converting a small number expressed in scientific notation into standard notation. The steps for converting a small number are the same as for a large number, but the decimal point should be shifted to the left instead of to the right.

Since scientific notation relies on powers of ten, it's simple to convert a number from scientific notation to standard notation or vice versa. To convert a large number (with a positive power) from scientific notation to standard notation, first identify the decimal point in the mantissa, then shift the decimal to the right by the number indicated by the power.

To convert a number from standard notation to scientific notation, just reverse these steps. Find the decimal point in the number, and move it until the number is at least 1 but less than 10. Count the number of places you moved the decimal point and use that number as the power. If you moved the decimal point to the left, make the power positive.

If you moved the decimal point to the right, make the power negative.

Scientific notation also makes it simpler to do multiplication and division. To multiply two numbers in scientific notation, multiply the mantissas and add the powers:

(3 × 104) × (4 × 10-2)
(3 × 4) × 104 – 2
12 × 102
1.2 × 103

In some cases, such as the one shown here, you may need to shift the decimal point again ensure that the number is in correct scientific notation. It should never be necessary to shift the decimal point by more than one digit. When dividing numbers in scientific notation, divide the mantissas and subtract the powers:

3 × 10 4 4 × 10 − 2 {displaystyle {frac {3 imes 10^{4}}{4 imes 10^{-2}}}}

( 3 / 4 ) × 10 4 + 2 {displaystyle (3/4) imes 10^{4+2}}

0.75 × 106
7.5 × 105

Here also, it may be necessary to shift the decimal point and change the exponent.

Scientific notation makes it easy to compare numbers that have very different values because all the zeroes have been replaced with the much more readable exponent. Numbers with a greater exponent are always bigger than numbers with a lesser exponent.

If one of the exponents is bigger than the other by more than a couple, the difference between the two is clearly very big.

Recognizing a huge difference between two numbers can sometimes be a very useful insight, so it often makes sense to take a moment to develop an intuitive feel for a math problem before attacking it. In some cases, it's useful to see roughly by how much one number is larger than another.

Scientific notation makes this much simpler. For a rough estimate, you only need to find the difference in the exponents. For example, 107 is greater than 103, since 7 – 3 = 4.

Some tourists in the Chicago Museum of Natural History are marveling at the dinosaur bones. One of them asks the guard, “Can you tell me how old the dinosaur bones are?”
The guard replies, “They are 73 million, four years, and six months old.”
“That's an awfully exact number,” says the tourist. “How do you know their age so precisely?”
The guard answers, “Well, the dinosaur bones were seventy three million years old when I started working here, and that was four and a half years ago.”
(From the Science Jokes Web page [1])

In science, measurements are never perfect and numbers are never exact. As a result, every measurement we make has some uncertainty associated with it. Scientific notation makes it easy to express how precisely a number is known. Suppose a paleontologist discovers ancient dinosaur bones and finds that they are 73 million years old.

Of course, the paleontologist doesn't know exactly how old they are. Maybe they're 73,124,987 years old, but the paleontologist only knows the age within 1 million years, so the age is written as 73,000,000 years, or 7.3 × 107 years.

Either of these expressions imply that the bones aren't exactly 73 million years old, but are 73 million years old, give or take a million years.

But what if the paleontologist knows the age within 200,000 years, and is sure that the bones aren't, say, 73.4 million years old? In that case, the standard notation is ambiguous — the number is still written as 73,000,000 years.

In scientific notation, we can write the number as 7.30 × 107 years. If we write this, we mean that the third digit is significant.

The paleontologist might have calculated that the bones are 72,954,332 years old, but it would be useless to report these numbers, since the error on this measurement was 200,000 years. The extra digits are insignificant.

The number of significant figures in a number are a reflection of the precision expressed in the number. In this case, the number of significant figures is three. The first significant figure is 7, the second is 3, and the third is 0.

## Scientific Notation

• Recognize how to convert between general and scientific notation
• Scientific notation is expressed in the form [latex]a imes 10^b[/latex] (where “b” is an integer and “a” is any real number), such as [latex]6.02 imes 10^{23}[/latex] .
• Scientific notation allows orders of magnitude to be more easily compared.
• E notation is another form of scientific notation, in which “E” replaces 10, such as 6.02 E 23. This number is the same as [latex]6.02 imes 10^{23}[/latex] .
• Basic operations are carried out in the same manner as with other exponential numbers.

Scientific notation is a more convenient way of writing very small or very large numbers.

The general representation for scientific notation is [latex]a imes 10^b[/latex](where “b” is an integer and “a” is any real number). When writing in scientific notation, only include significant figures in the real number, “a.” Significant figures are covered in another section.

To express a number in scientific notation, you move the decimal place to the right if the number is less than zero or to the left if the number is greater than zero.For example, in 456000, the decimal is after the last zero, so to express this in scientific notation, you would need to move the decimal to in between the 4 and 5.

The decimal would move five places to the left to get 4.56 as our [latex]a[/latex] in [latex]a imes 10^b[/latex]. The number of times you move the decimal place becomes the integer “b.

” In this case, the decimal moved five times. Therefore, our number in scientific notation would be: [latex]4.56 imes 10^5[/latex].

Keep in mind that zeroes are not included in “a” because they are not significant figures.

In order to go between scientific notation and decimals, the decimal point is moved the number of spaces indicated by the exponent. A negative exponent tells you to move the decimal point to the right, while a positive exponent tells you to move it to the left.