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# Scientific Notation

Have you wondered what the distance between the Earth and the moon was? Can you guess what it might be in meters? Well, based on a study made by NASA, the distance spans approximately 382,500,000 meters. I don't know about you, but that's quite a hefty lot of numbers for me to handle. However, not to worry. This is where scientific notation comes in! It provides us with a method to deal with extremely large numbers or significantly small ones in an easier way.

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. $$x$$

## 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

$a×{10}^{b}$

where $1<=|a|<10$ 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

$a×{10}^{b}$

where $1<=|a|<10$, 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 $2$ $2×{10}^{0}$ $a=2,b=0$ $20$ $2×{10}^{1}$ $a=2,b=1$ $200$ $2×{10}^{2}$ $a=2,b=2$

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 $0.2$ $2×{10}^{-1}$ $a=2,b=-1$ $0.02$ $2×{10}^{-2}$ $a=2,b=-2$ $0.002$ $2×{10}^{-3}$ $a=2,b=-3$

## 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 $7×{10}^{2}$. The number $7×{10}^{2}$ then is the scientific notation of 700.

The format for scientific notation is $a×{10}^{b}$ 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 mantissa is the part of a logarithm after the decimal point.

## Examples of Scientific Notation

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

1. $650,000,000=6.5×{10}^{8}$
2. $75=7.5×{10}^{1}=7.5×10$
3. $5.05×{10}^{7}$
4. $0.00001=1×{10}^{-5}$
5. $1230000000=1.23×{10}^{9}$

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.

$76400=76.4×{10}^{3}$

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

$76400=7.64×{10}^{4}$

Here is another worked example.

$160=2.5×{8}^{2}$

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

$160=1.6×{10}^{2}$

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

$0.034=34×{10}^{-3}$

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

$0.034=3.4×{10}^{-2}$

## 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

$7×{10}^{9}$.

#### 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

$0.0024=2.4×{10}^{-3}$.

### 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 $3.825×{10}^{8}$ 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 $4.0×{10}^{-7}$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.

1. 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.

2. Add or subtract these decimal numbers.

3. Write your number in scientific notation if necessary.

Here is an example that demonstrates this.

$\left(6.7×{10}^{4}\right)+\left(5.87×{10}^{5}\right)$.

Rewriting the number will have us with

$\left(0.67×{10}^{5}\right)+\left(5.87×{10}^{5}\right)$

We will now have;

$\left(0.67+5.87\right)×{10}^{5}$.

With our example, we will have;

$6.54×{10}^{5}$

#### 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 $2.33×{10}^{8}$meters from her home to her workplace. After work, she was told to attend a meeting in town. From her workplace, she travels $8.2×{10}^{9}$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 $2.33×{10}^{8}$ and $8.2×{10}^{9}$ since we are looking for her total journey. In doing so, we can write this as

$\left(2.33×{10}^{8}\right)+\left(8.2×{10}^{9}\right)$

Rewrite both to have the same exponent.

$\left(0.233×{10}^{9}\right)+\left(8.2×{10}^{9}\right)$

$\left(0.233+8.2\right)×{10}^{9}$

Thus, we have $8.433×{10}^{9}$ meters in total.

### Multiplying and Dividing Scientific Notation

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

1. Multiply or divide the decimal numbers.

2. Now either multiply by adding exponents or divide by subtracting the exponents of the 10.

Here is a worked example.

Solving $\left(8.4×{10}^{-3}\right)\left(6.1×{10}^{6}\right)$, we will have

$8.4×6.1=51.24$

Then,

${10}^{-3}×{10}^{6}={10}^{-3+6}={10}^{3}$

We will now have

$51.24×{10}^{3}$

#### 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 $6×{10}^{7}$ , and its length to be $8×{10}^{5}$, find its width.

Solution

If $Perimeter=Length×Width$ then rearranging this will become

$Width=\frac{Perimeter}{Length}$

Divide the decimal numbers.

$6÷8=0.75$

Subtract exponents of 10.

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

$0.75×{10}^{2}$

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.

$Width=7.5×{10}^{1}cm$

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 $a×{10}^{b}$ where $1<=|a|<10$ 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.

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

• 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.

The scientific notation for 700 is 7 ✕ 10^2

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 ✕ 10 ^ 8.

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.

## Final Scientific Notation Quiz

Question

A number can be said to be written in scientific notation when a number between 1 and 10 is multiplied by a power of .....

10

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Question

The exponent of 10 in scientific notation is an integer.

False

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