Lesson Explainer: Scientific Notation | Nagwa Lesson Explainer: Scientific Notation | Nagwa

# Lesson Explainer: Scientific Notation Mathematics • First Year of Preparatory School

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In this explainer, we will learn how to express numbers in scientific notation and how to convert numbers between their normal and scientific forms.

When numbers have a very large or very small absolute value, writing them implies writing a lot of digits, for instance, 23‎ ‎410‎ ‎000‎ ‎000 or 0.0000016482. This is not very convenient. Therefore, we use another way of writing numbers called scientific notation (also referred to as standard form). It is a compact way of writing numbers that proves particularly useful for numbers with a very large or very small absolute value.

To explain scientific notation, let us consider a big number: 300‎ ‎000‎ ‎000. We note that this number is a 3 followed by 8 zeros, and ideally, we want to be able to express this without having to write all of the zeros down. We note that one way to write a number with 8 zeros succinctly is to write it in powers of 10:

Since the number we are interested in is three times this number, we can write it as

In this way, we have written the number purely in terms of an exponent of 10 and a number multiplying it, meaning it is much quicker to write.

We can extend this approach to very small values in a similar way. For instance, 0.0000002 can be written in scientific notation as follows:

We can see that the main difference with this form compared with the larger number is that the power of 10 is now negative. Generally speaking, larger numbers can be written in terms of positive powers of 10, while smaller numbers can be written using negative powers.

Let us consider how this approach can be generalized with the following definition.

### Definition: Scientific Notation

A number written in scientific notation (also called standard form) is of the form where .

The exponent is positive for numbers with a large absolute value. It is negative for numbers with a very small absolute value.

Let us quickly check our understanding of this definition with the first example.

### Example 1: Identifying When a Number Is Correctly Written in Scientific Notation

Which of the following numbers is not in standard form?

Recall that a number written in scientific notation (also called standard form) is of the form , where .

The four options listed here are all in the form of a decimal or integer number multiplied by , with either positive or negative. However, in option A, this number is the decimal , where , which is smaller than 1.

Therefore, the number is not written in standard form.

When converting a number from its normal form to scientific notation and vice versa, it is important to understand what happens when a number is divided or multiplied by 10 or powers of 10.

Recall that, in our decimal system, each digit has a value that can be written in the form of a power of ten. In a place value table, ones can indeed be replaced with , tens with , hundreds with , and so on, and as for the digits after the decimal point, tenths are , hundredths are , thousandths are , and so forth.

Multiplying a number by 10 is therefore equivalent to moving all its digits one place left (a one becomes a ten, a ten becomes a hundred, a tenth becomes a one, etc.), while dividing by 10 is equivalent to moving all the digits one place right (a one becomes a tenth, a ten becomes a one, a tenth becomes a hundredth, etc.). Look, for instance, at 347.58, , and .

It follows that multiplying by 100, or , is equivalent to moving all the digits two places left, while dividing by is equivalent to moving them two places right, and so forth. Remember that dividing by is the same as multiplying by since .

These translations of the digits to the left or to the right when multiplying by a power of ten can also be tracked by the movement of the decimal point. When a number is multiplied by 10, its decimal point moves one place right. When a number is divided by 10, it moves one place left.

When a number is written in scientific notation, we can easily express it in normal form by carrying out multiplication by the power of 10 that appears in its scientific notation form. Let us look at a couple of examples to see how this works; the first involves a large number.

### Example 2: Expressing a Large Number in Normal Form

Express in normal form.

Recall that a number written in scientific notation is of the form , where .

Note that the given number is already written in scientific notation. To express in normal form, we need to multiply 3.06707 by .

Multiplying 3.06707 by 10 moves the decimal point 1 place right to give 30.6707, multiplying it by 100 moves the decimal point 2 places right to give 306.707, multiplying it by 1‎ ‎000 gives 3‎ ‎067.07, and multiplying it by 10‎ ‎000 gives 30‎ ‎670.7.

However, once we get to multiplying 3.06707 by 100‎ ‎000, moving the decimal point 5 places right means that it effectively reaches the end of the digits of our original number, giving 306‎ ‎707. The way to go beyond this, enabling us to multiply by further powers of 10, is to add as many trailing zeros to our original number as we need. Note that we can do this without changing its value, because 3.06707 has the same numerical value as 3.067070, 3.0670700, or (as needed here) 3.0670700000.

Therefore, multiplying 3.06707 by moves the decimal point 10 places to the right, so we get 30‎ ‎670‎ ‎700‎ ‎000.

Now, we work through the same method for a small number.

### Example 3: Expressing a Small Number in Normal Form

Write in normal form.

Recall that a number written in scientific notation is of the form , where .

The given number is already written in scientific notation; to express in normal form, we need to multiply 3.01 by .

Multiplying 3.01 by is equivalent to dividing by 10, which moves the decimal point 1 place left. However, this means the decimal point effectively reaches the start of the digits of our original number, so we need to add a leading zero, giving the number 0.301.

The way to go beyond this, enabling us to divide by further powers of 10, is to add as many leading zeros to our original number as we need. Note that we can do this without changing its value, because 3.01 has the same numerical value as 03.01, 003.01, or (as needed here) 000‎ ‎003.01.

Therefore, multiplying 3.01 by moves the decimal point 5 places to the left, so we get 0.0000301.

Next, we will learn what to do when going in the opposite direction. We want to be able to write any number in scientific notation.

Let us start with the very large number 23‎ ‎410‎ ‎000‎ ‎000. For this, we are going to write 23‎ ‎410‎ ‎000‎ ‎000 in a place value table.

 10 10 10 10 10 10 10 10 10 10 10 2 3 4 1 0 0 0 0 0 0 0

The number 23‎ ‎410‎ ‎000‎ ‎000 can be decomposed as the sum of and

We see that the digit with the highest value is the first digit on the left, here 2, and its value is . Therefore, .

Note that can be decomposed as , and we have indeed , , and .

Now, we look at a very small number. Choosing 0.0000016482, again we start by writing it in a place value table.

 10 10 10 10 10 10 10 10 10 10 10 0 0 0 0 0 0 1 6 4 8 2

We can decompose here as well 0.0000016482 as the sum of and

The digit with the highest value in 0.0000016482 is the first nonzero digit after the decimal place, here 1, and its value is . Therefore, .

Note that we have indeed , , , and .

We see that writing a number in a place value table is a very effective method to write the number in scientific notation. However, it is a little bit long. A faster method is to compare the position of the decimal point in the original, normal number with that in the number from its scientific notation version . Note that when given any normal number, we can always write down the corresponding value of . This is because there will be only one placement of the decimal point that results in a number with an absolute value that is at least 1 and less than 10. The value of is then the number of places the decimal point needs to move from its position in to its position in the original, normal number. If the decimal point needs to move right, then is positive because is smaller than the number, and so must be multiplied by a power of 10. If the decimal point needs to move left, then is negative because is larger than the number and so must be divided by a power of 10, that is, multiplied by a power of 10 with a negative exponent.

To test our knowledge of this method, let us go through a few examples. The first one concerns finding a missing value of when a normal number is expressed in its scientific notation form .

### Example 4: Finding an Unknown by Comparing Numbers in Scientific Notation

Given that , find the value of .

Recall that a number written in scientific notation is of the form , where .

Here, we have a number written in its normal form, together with the value of when it is written in scientific notation. We need to work out the value of , the missing exponent in the scientific notation form .

The number tells us how many times 4.16 needs to be multiplied or divided by 10 to give 4‎ ‎160‎ ‎000. Here, we see that 4.16 is smaller than 4‎ ‎160‎ ‎000, so 4.16 must be multiplied by 10 several times.

To find the value of , we count how many places the decimal point in 4.16 would need to move right to its position in 4‎ ‎160‎ ‎000. We find that it is six times.

Therefore, , so .

Next, we will practice writing a large number in scientific notation form.

### Example 5: Expressing a Large Number in Scientific Notation

Express 874‎ ‎527‎ ‎893 in scientific notation.

Recall that a number written in scientific notation is of the form , where .

To express 874‎ ‎527‎ ‎893 in scientific notation, we need first to determine the value of .

The number is made with exactly the same digits of 874‎ ‎527‎ ‎893 and in the same order, but its absolute value must be greater than or equal to 1 and less than 10. Hence, .

Then, we need to find the value of . The number tells us how many times needs to be multiplied or divided by 10 to give 874‎ ‎527‎ ‎893. Here, we see that 8.74527893 is smaller than 874‎ ‎527‎ ‎893. Therefore, needs to be multiplied several times by 10.

To easily find how many times the multiplication by 10 needs to be repeated, we find by how many places the decimal point in 8.74527893 would need to move right to its position in 874‎ ‎527‎ ‎893 upon this repeated multiplication by 10. We find that it is eight times.

Therefore, , and so if we express 874‎ ‎527‎ ‎893 in scientific notation, we get .

We follow this with an example of what happens for a small number.

### Example 6: Expressing a Small Number in Scientific Notation

Express 0.00447 in scientific notation.

Recall that a number written in scientific notation is of the form , where .

To express 0.00447 in scientific notation, we need first to determine the value of .

The number is made with exactly the same digits of 0.00447 and in the same order, but its absolute value must be greater than or equal to 1 and less than 10. Hence, . The zeros to the left of the first 4 do not need to be written.

Then, we need to find the value of . The number tells us how many times needs to be multiplied or divided by 10 to give 0.00447. Here, we see that 4.47 is larger than 0.00447. Therefore, needs to be divided several times by 10. We are going to write this repeated division by 10 in the form of a repeated multiplication by . In other words, is negative and its absolute value tells us how many times the multiplication by needs to be repeated.

To easily find the absolute value of , we find by how many places the decimal point in 4.47 would need to move left to its position in 0.00447 upon this repeated multiplication by . We find that it is three times.

Therefore, , and so if we express 0.00447 in scientific notation, we get .

Since many quantities that feature in real-world situations can be either very large (e.g., the area of a country in square kilometres) or very small (e.g., the mass of an atom in grams), then scientific notation is extremely useful in helping us to describe and compare the numbers involved. Let us finish with an example of this type, which features large numbers.

### Example 7: Ordering Numbers from Least to Greatest Involving Scientific Notation in a Real-World Context

The table shows the populations of five countries in a given year. List the countries from the smallest to the largest population.

 Country Population A B C D E 5.3×10 6.1×10 5.8×10 6.5×10 4.2×10

Here, we are given the population numbers of five countries written in scientific notation. We need to compare these numbers so that we can list the countries from the smallest to the largest population.

Recall that a number written in scientific notation is of the form , where . To answer this question, it is important to understand the following two facts about numbers written in scientific notation:

1. For any two numbers that have different exponents of 10, the larger number will be the one with the larger exponent.
2. For any two numbers that have the same exponent of 10, the larger number will be the one with the larger value of .

In this question, we have five population numbers with five different exponents of 10, namely 4, 5, 6, 7, and 8. Point I above tells us that the larger the exponent, the larger the number. This means that we can rank the populations from smallest to largest without doing any detailed calculations.

The country with the smallest population is C with a population of . Next comes country B with a population of , followed by D with a population of , E with a population of , and A with a population of .

Therefore, listing the countries from the smallest to the largest population gives C, B, D, E, and A.

Let us finish by recapping some key concepts from this explainer.

### Key Points

• A number written in scientific notation (also referred to as standard form) is of the form where . The exponent is positive for numbers with a large absolute value; it is negative for numbers with a very small absolute value.
• To write a number in scientific notation, we first find , that is, the number with exactly the same digits as our number, but with . Then, we find the value of , that is, the number of places the decimal point needs to move from its position in to its position in the original number.
If (i.e., if the decimal point needs to move to the right from to the number), then is positive. If (i.e., if the decimal point needs to move to the left from to the number), then is negative.
• When a number is written in scientific notation, we can express it in its normal form by carrying out the multiplication by the appropriate power of 10.
• For any two numbers written in scientific notation, if they have different exponents of 10, then the larger number will be the one with the larger exponent. If they have the same exponent of 10, then the larger number will be the one with the larger value of .

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