In this article, we will discuss the concept of a scientific notation. Furthermore, we will become acquainted with a technique that demonstrates how we can convert a number from standard form to its corresponding scientific notation and vice versa.

## What is a Scientific Notation?

To begin our topic, let us first define the meaning of a scientific notation.

**Scientific notation**, also known as standard form, is a method that expresses (or rewrites) a multi-digit number in a compact way. It takes the form

$$a\times 10^b$$

where \(1\leq |a|< 10\) and \(b\) is an integer.

This is a very effective way to write very large numbers or very small ones as well. It may be helpful to note that for the structure introduced above, that is

$$a\times 10^b$$

where \(1\leq |a|< 10\), \(a\) is the coefficient and \(10\) is a constant base. The table below shows several examples where scientific notation takes place.

Number | Scientific notation | a and b values |

\(2\) | \(2\times 10^0\) | \(a=2, b=0\) |

\(20\) | \(2\times 10^1\) | \(a=2,b=1\) |

\(200\) | \(2\times 10^2\) | \(a=2,b=2\) |

What this means is that a larger number can be rewritten in a shorter manner by increasing the power of \(10\). Or in other words, the value of \(b\).

Coincidentally, this works in the reverse too. Numbers that are particularly close to zero can also be expressed in this way by changing the sign of the exponent. Here is a table that demonstrates a few examples.

Number | Scientific notation | \(a\) and \(b\) values |

\(0.2\) | \(2\times 10^{-1}\) | \(a=2,b=-1\) |

\(0.02\) | \(2\times 10^{-2}\) | \(a=2,b=-2\) |

\(0.002\) | \(2\times 10^{-3}\) | \(a=2,b=-3\) |

## Writing Numbers in Scientific Notation

A number can be written in scientific notation by expressing the value as **a number between \(1\) and \(10\) multiplied by a power of \(10\)**. For example, the number \(700\) can be written as \(7\times 10^2\). The number \(7\times 10^2\) then is the scientific notation of \(700\).

The format for scientific notation is \(a\times 10^b\) where

**\(a\)**is a number or decimal number such that the absolute value of \(**a\)**is**greater than**or**equal**to**one**and**less than ten**and;**\(b\)**is the**power of \(10\)**required so that the scientific notation is mathematically equivalent to the original number.

### Rules for Writing Numbers in Scientific Notation

To write a number in scientific notation, one must adhere to the following rules:

The base is always \(10\).

The value of the coefficient is always greater or equal to \(1\) and less than \(10\).

The coefficients can also be either positive or negative values.

The rest of the significant digits of the number is carried by the mantissa.

The

is the part of a logarithm after the decimal point.*mantissa*

## Examples of Scientific Notation

In this section, we will take a look at a worked example involving scientific notation.

- \(650,000,000=6.5\times 10^8\)
- \(75=7.5\times 10^1=7.5\times 10\)
- \(5.05\times 10^7\)
- \(0.00001=1\times 10^{-5}\)
- \(1,230,000,000=1.23\times 10^9\)

Notice that all the conditions for writing scientific notations are met in the example above:

- the base of each example is \(10\);
- the coefficients are greater or equal to \(1\) and less than \(10\);
- and the exponent of the base is accounted for the mantissa.

### Common Errors in Scientific Notation

Although the method behind scientific notation can be straightforward, there are still some common errors you should consider so that you don't fall trap of making careless mistakes in your work. Here is an example where scientific notations are not valid.

\(76400=76.4\times 10^3\).

This is incorrect because the**coefficient needs to be between \(1\) and \(10\)**. In this case, it should be \(7.64\), not \(76.4\). The correct answer would be

$$76400=7.64\times 10^4$$

Here is another worked example.

\(160=2.5\times 8^2\)

Although this equation may be true, it is not a valid scientific notation. Notice the base used in the example above. Recall that**all scientific notations possess a base of \(10\)**. The base in this situation is \(8\). Thus, the correct answer should be

$$160=1.6\times 10^2$$

Let's look at one more example before we move on to our next section.

\(0.034=34\times 10^{-3}\).

As before,**the coefficient needs to be between \(1\) and \(10\)**. In this case, \(34\) is indeed more than \(10\). The correct scientific notation would be

$$0.034=3.4\times 10^{-2}$$

## Standard Form and Scientific Notation

In this section, we will be learning how to interchange a given number between its standard form and scientific notation.

### Converting Numbers in Standard Form to Scientific Notation

To understand how we can convert numbers in standard form to their appropriate scientific notation, we shall demonstrate two cases for you to consider.

#### Case 1: The decimal point moves to the left if the given number is more than \(10\).

This means that the power of \(10\) here will be a positive value. Here is an example.

The population of the world is currently at \(7,000,000,000\). To express this in scientific notation, we can write this as

$$7\times 10^9$$

#### Case 2: The decimal point moves to the right if the given number is less than \(1\).

In this instance, the power of \(10\) will become a negative value. Here is an example.

If we have to write the diameter of a grain of sand, which is \(24\) ten-thousandths inches or \(.0024\) inch, we would get

$$0.0024=2.4\times 10^{-3}$$

### Converting a Number in Scientific Notation to Standard Form

There is no particular rule to follow when converting a number in scientific notation to standard form. However, there are a few pointers we should note when doing so.

Move the decimal point to the

**right**if the exponent of the base is**positive**.Move the decimal point to the

**left**if the exponent of the base is**negative**.Move the decimal point

**as many times**as indicated by the**exponent**.In standard form, do not write multiply by 10 anymore.

Let us take a look at a few examples to see how this is done.

The scientific notation of the distance from the Earth to the moon is \(3.825\times 10^8\) meters. How would you represent this number in standard form?

**Solution**

Since the exponent of the base is positive, move the decimal point to the right. By the third movement, we should be at \(3825\). This means any movement after this adds a \(0\) to the figure. This will add \(5\) more zeros to the number. Thus resulting in \(382,500,000\) meters.

Here is another example.

The length of the shortest wavelength of visible light is considered to be \(4.0\times 10^{-7}\) meters. Write this in standard form.

**Solution**

The decimal point will be moved to the left since the exponent of the base is negative. One move of the decimal point will leave us at \(0.4\). However, we need to do all \(7\) movements. Each one after that will add a zero before \(4\). Hence, we will have \(0.0000004\) meters.

## Arithmetic Operations with Scientific Notation

In this segment, we will discuss how we can perform basic arithmetic operations with numbers in scientific notation. It can get rather complex and confusing when dealing with extremely large or significantly small numbers. The purpose of scientific notation is to make numbers easier to read, write, and calculate. Numbers in scientific notation can be added, subtracted, multiplied, and divided while they are still in scientific notation.

### Adding and Subtracting Numbers in Scientific Notation

Below are steps to add and subtract numbers in scientific notation.

Make both numbers you are attempting to add or subtract have

**the same exponent**by rewriting the number with the smaller exponent and moving the decimal point to its decimal number the required number of times.Add or subtract these decimal numbers.

Write your number in scientific notation if necessary.

Here is an example that demonstrates this.

\((6.7\times 10^4)+(5.87\times 10^5)\).

Rewriting the number will have us with

$$(0.67\times 10^5)+(5.87\times 10^5)$$

We will now have;$$(0.67+5.87)\times 10^5$$

With our example, we will have;

$$6.54\times 10^5$$

#### Real-world Example Involving Addition and Subtraction of Scientific Notation

In this section, we will observe a real-world problem that makes use of adding and subtracting numbers in scientific notation.

Amy travels \(2.33\times 10^8\) meters from her home to her workplace. After work, she was told to attend a meeting in town. From her workplace, she travels \(8.2\times 10^9\) meters to the venue of this meeting. Find the total distance she covered today.

**Solution**

This problem is asking you to add the two given numbers in scientific notation, namely \(2.33\times 10^8\) and \(8.2\times 10^9\) since we are looking for her total journey. In doing so, we can write this as

$$(2.33\times 10^8)+(8.2\times 10^9)$$

Rewrite both to have the same exponent.

$$(0.233\times 10^9)+(8.2\times 10^9)$$

Add the decimals

$$(0.233+8.2)\times 10^9$$

Thus, we have \(8.433\times 10^9\) meters in total.

### Multiplying and Dividing Scientific Notation

Below are steps to multiply or divide numbers in scientific notation.

Multiply or divide the decimal numbers.

Now either multiply by adding exponents or divide by subtracting the exponents of the \(10\).

Write your answer in scientific notation if necessary.

Here is a worked example.

Solving \((8.4\times 10^{-3})(6.1\times 10^6)\), we will have

$$8.4\times 6.1=51.24$$

Then,

$$10^{-3}\times 10^6=10^{-3+6}=10^3$$

We will now have

$$51.24\times 10^3$$

#### Real-world Example Involving Multiplication and Division of Scientific Notation

In this section, we will observe a real-world problem that makes use of multiplying and dividing numbers in scientific notation.

Given the perimeter of a rectangle to be \(6\times 10^7\), and its length to be \(8\times 10^5\), find its width.

**Solution**

If \(\text{Perimeter}=\text{Length}\times\text{Width}\) then rearranging this will become

$$\text{Width}=\dfrac{\text{Perimeter}}{\text{Length}}$$

Divide the decimal numbers.

$$6\div 8=0.75$$

Subtract exponents of \(10\).

$$10^{7-5}=10^2$$

$$0.75\times 10^2$$

Following the rules of scientific notation, the coefficient needs to be between \(1\) and \(10\). Hence, this can be worked on more by moving the decimal point to the right in 1 decimal place. Moving the decimal point to the right by 1 reduces the exponent of its base by 1 also.

$$\text{Width}=7.5\times 10^1\text{ cm}$$

Multiplying scientific notation is finding the product of their coefficients and adding their exponents. In this regard, dividing them is also equivalent to finding their quotient and subtracting their exponents.

## Scientific Notation - Key takeaways

- Scientific notation is a way to rewrite multi-digit numbers in a compact way in the form \(a\times 10^b\) where \(1\leq |a|<10\) and \(b\) is an integer.
- The value of the coefficient is always greater or equal to \(1\) and less than \(10\).
- The base in scientific notation is always \(10\).
- When adding or subtracting numbers in scientific notation, be sure all exponents involved have the same value.
- When multiplying in scientific notation, multiply the coefficients and add the exponents of the base.
- When dividing in scientific notation, divide the coefficients and subtract the exponents of the base.

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##### Frequently Asked Questions about Scientific Notation

What is scientific notation?

Scientific notation is a symbolic system of representation of exceptionally complex numbers.

What are the rules in scientific notation?

- The base is always 10.
- The exponent can be a negative or positive value but not 0.
- The value of the coefficient is always greater or equal to 1 and less than 10.
- The coefficients can also be either positive or negative values.
- The rest of the significant digits of the number is carried by the mantissa.

What is an example of scientific notation?

The scientific notation for 700 is 7 x 10^{2}.

How do you write scientific notation?

A number is written in scientific notation when a number between 1 and 10 is multiplied by a power of 10. For example, 650,000,000 can be written in scientific notation as 6.5 x 10^{8}.

How to use multiplication and division in scientific notation?

Multiply or divide the decimal numbers then you multiply the exponents if you are multiplying, and subtract exponents if you're dividing. The answer is then written in scientific notation.

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