Explainer: Scientific Notation

In this explainer, we will learn how to express numbers in scientific notation and to convert numbers between their standard 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 the scientific notation. It is a compact way of writing numbers that proves particularly useful for numbers with a very large or very small absolute value.

The idea of scientific notation is to write numbers in the form ๐‘Žร—10๏Š, where 1โ‰ค|๐‘Ž|<10. For numbers with a very large absolute value, ๐‘› is positive. For instance, 300,000,000=3ร—10๏Šฎ. And for numbers with a very small absolute value, ๐‘› is negative. For instance, 0.0000002=2ร—10๏Šฑ๏Šญ.

Definition: Scientific Notation

A number written in scientific notation is written in the form ๐‘Žร—10,๏Š where 1โ‰ค|๐‘Ž|<10.

The exponent ๐‘› 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 scientific notation?

  1. โˆ’0.57ร—10๏Šฑ๏Šซ
  2. 9.1ร—10๏Šฏ
  3. 5ร—10๏Šฑ๏Šฏ
  4. 7ร—10๏Šฎ

Answer

The four numbers given are all in the form ๐‘Žร—10๏Š, with ๐‘› either positive or negative.

However, for one of them, |๐‘Ž|=0.57, and 0.57 is smaller than 1. Therefore, โˆ’0.57ร—10๏Šฑ๏Šซ is not written in scientific notation.

When converting a number from the standard 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 10๏Šฆ, tens with 10๏Šง, hundreds with 10๏Šจ, and so on, and as for the digits after the decimal point, tenths are 10๏Šฑ๏Šง, hundredths are 10๏Šฑ๏Šจ, thousandths are 10๏Šฑ๏Šฉ, 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, 347.58ร—10=3,475.8, and 347.58รท10=34.758.

It follows that multiplying by 100, or 10๏Šจ, is equivalent to moving all the digits two places left, while dividing by 10๏Šจ is equivalent to moving them two places right, and so forth. Remember that dividing by 10๏Šจ is the same as multiplying by 10๏Šฑ๏Šจ since 10=110๏Šฑ๏Šจ๏Šจ.

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 standard form by carrying out multiplication by the power of 10. Let us look at a couple of examples to see how it works.

Example 2: Expressing a Large Number in Standard Form

Express 3.06707ร—10๏Šง๏Šฆ in standard form.

Answer

To express 3.06707ร—10๏Šง๏Šฆ in standard form, we need to multiply 3.06707 by 10๏Šง๏Šฆ.

Multiplying 3.06707 by 10 gives 30.6707, multiplying it by 100 gives 306.707, and so on. So multiplying by 10๏Šง๏Šฆ moves the decimal point in 3.06707โ€‰โ€‰10 places right. Hence, we get 30,670,700,000.

Example 3: Expressing a Small Number in Standard Form

Write 3.01ร—10๏Šฑ๏Šซ in standard form.

Answer

To express 3.01ร—10๏Šฑ๏Šซ in standard form, we need to multiply 3.01 by 10๏Šฑ๏Šซ. Multiplying 3.01 by 10๏Šฑ๏Šง is equivalent to dividing by 10 and it gives 0.301, multiplying it by 10๏Šฑ๏Šจ gives 0.0301, and so on. So multiplying by 10๏Šฑ๏Šซ moves the decimal point in 3.01 five places left. Hence, we get 0.0000301.

Now, we want to be able to write any number in scientific notation. Let us start with 23,410,000,000. For this, we are going to write 23,410,000,000 in a place value table.

10๏Šง๏Šฆ10๏Šฏ10๏Šฎ10๏Šญ10๏Šฌ10๏Šซ10๏Šช10๏Šฉ10๏Šจ10๏Šง10๏Šฆ
23410000000

The number 23,410,000,000 can be decomposed as the sum of 2ร—10,3ร—10,4ร—10,๏Šง๏Šฆ๏Šฏ๏Šฎ and 1ร—10.๏Šญ

We see that the digit with the highest value is the first digit on the left, here 2, and its value is 10๏Šง๏Šฆ. Therefore, 23,410,000,000=2.341ร—10๏Šง๏Šฆ.

Note that 2.341ร—10๏Šง๏Šฆ can be decomposed as 2ร—10+0.3ร—10+0.04ร—10+0.001ร—10๏Šง๏Šฆ๏Šง๏Šฆ๏Šง๏Šฆ๏Šง๏Šฆ, and we have indeed 0.3ร—10=3ร—10๏Šง๏Šฆ๏Šฏ, 0.04ร—10=4ร—10๏Šง๏Šฆ๏Šฎ, and 0.001ร—10=1ร—10๏Šง๏Šฆ๏Šญ.

Let us write now 0.0000016482 in a place value table.

10๏Šฆ10๏Šฑ๏Šง10๏Šฑ๏Šจ10๏Šฑ๏Šฉ10๏Šฑ๏Šช10๏Šฑ๏Šซ10๏Šฑ๏Šฌ10๏Šฑ๏Šญ10๏Šฑ๏Šฎ10๏Šฑ๏Šฏ10๏Šฑ๏Šง๏Šฆ
00000016482

We can decompose here as well 0.0000016482 as the sum of 1ร—10,6ร—10,4ร—10,8ร—10,๏Šฑ๏Šฌ๏Šฑ๏Šญ๏Šฑ๏Šฎ๏Šฑ๏Šฏ and 2ร—10.๏Šฑ๏Šง๏Šฆ

The digit with the highest value in 0.0000016482 is the first nonzero digit after the decimal place, here 1, and its value is 10๏Šฑ๏Šฌ. Therefore, 0.0000016482=1.6482ร—10๏Šฑ๏Šฌ.

Note that we have indeed 0.6ร—10=6ร—10๏Šฑ๏Šฌ๏Šฑ๏Šญ, 0.04ร—10=4ร—10๏Šฑ๏Šฌ๏Šฑ๏Šฎ, 0.008ร—10=8ร—10๏Šฑ๏Šฌ๏Šฑ๏Šฏ, and 0.0002ร—10=2ร—10๏Šฑ๏Šฌ๏Šฑ๏Šง๏Šฆ.

We see that writing a number in a place value table is a very effective method to write a number in scientific notation. However, it is a little bit long. A faster method is to look at the position of the decimal point in the original number and in ๐‘Ž once the number is written in scientific notation as ๐‘Žร—10๏Š. The value of ๐‘› is then the number of places the decimal point needs to move from its position in ๐‘Ž to its position in the original number. If it needs to move right, then ๐‘› is positive because ๐‘Ž is smaller than the number, and so ๐‘Ž needs to be multiplied by a power of ten; if it needs to move left, then ๐‘› is negative because ๐‘Ž is larger than the number and needs to be divided by a power of 10, that is, to be multiplied by a power of 10 with a negative exponent.

Now, let us go through a couple of examples.

Example 4: Expressing a Large Number in Scientific Notation

Express 874,527,893 in scientific notation.

Answer

To express 874,527,893 in scientific notation, that is, in the form ๐‘Žร—10๏Š, we need first to determine the value of ๐‘Ž.

The number ๐‘Ž 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, ๐‘Ž=8.74527893.

Then, we need to find the value of ๐‘›. The number ๐‘› tells us how many times ๐‘Ž 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, ๐‘Ž 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.

Hence, ๐‘›=8, and 874,527,893=8.74527893ร—10๏Šฎ.

Example 5: Expressing a Small Number in Scientific Notation

Express 0.00447 in scientific notation.

Answer

To express 0.00447 in scientific notation, that is, in the form ๐‘Žร—10๏Š, we need first to determine the value of ๐‘Ž.

The number ๐‘Ž 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, ๐‘Ž=4.47. The zeros to the left of the first 4 do not need to be written.

Then, we need to find the value of ๐‘›. The number ๐‘› tells us how many times ๐‘Ž needs to be multiplied or divided by 10 to give 0.00447. Here, we see that 4.47 is larger than 0.00447. Therefore, ๐‘Ž 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 10๏Šฑ๏Šง. In other words, ๐‘› is negative and its absolute value tells us how many times the multiplication by 10๏Šฑ๏Šง needs to be repeated.

To easily find the absolute value of ๐‘›, 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 10๏Šฑ๏Šง. We find that it is three times.

Hence, ๐‘›=โˆ’3, and 0.00447=4.47ร—10๏Šฑ๏Šฉ.

Key Points

  1. A number written in scientific notation is written in the form ๐‘Žร—10,๏Š where 1โ‰ค|๐‘Ž|<10. The exponent ๐‘› is positive for numbers with a large absolute value. It is negative for numbers with a very small absolute value.
  2. To write a number in scientific form, we first find ๐‘Ž, that is, the number with exactly the same digits as our number but so that 1โ‰ค|๐‘Ž|<10. Then, we find the value of ๐‘›, that is, the number of places the decimal point needs to move from its position in ๐‘Ž to its position in the original number.
    If ||>|๐‘Ž|number (the decimal points need to move to the right from ๐‘Ž to the number), then ๐‘› is positive.
    If ||<|๐‘Ž|number (the decimal points need to move to the left from ๐‘Ž to the number), then ๐‘› is negative.
  3. When a number is written in scientific notation, we can express it in standard form by carrying out the multiplication by the power of 10.

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