# Video: Scientific Notation

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

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### Video Transcript

In this video, we will learn how to express numbers in scientific notation, sometimes referred to as standard form. We will look at how we can write very large and very small numbers in this notation. When numbers have a very large or very small absolute value, writing them means writing a lot of digits, for example, 241300000. At the other extreme, we have 0.000000764. Scientific notation is a compact way of writing numbers of this form.

Let’s begin by looking at a definition of scientific notation. A number written in scientific notation is written in the form 𝑎 multiplied by 10 to the power of 𝑏, where the absolute value of 𝑎 is less than 10 and greater than or equal to one. This means that 𝑎 could take any of the following values: four, seven, 3.2, negative 6.3, etcetera. The exponent, or power, 𝑏 is a positive integer for large numbers and a negative integer for small numbers.

The following are examples of numbers written in scientific notation. Four multiplied by 10 to the power of seven, 3.2 multiplied by 10 to the power of eight, and 2.14 multiplied by 10 to the power of negative seven. 0.6 multiplied by 10 to the power of eight and 23 multiplied by 10 to the power of negative five are not in scientific notation. This is because their value of 𝑎 is either less than one or greater than or equal to 10. 23 is greater than 10, and 0.6 is less than one. We will now look at some questions that involve expressing numbers in scientific notation.

There are approximately 200 million cows in India. Express this number in scientific notation.

Let’s firstly consider how we would write the number 200 million. The number one million is written as a one followed by six zeros. This means that 200 million can be written as 200 followed by another six zeros. Altogether, we have two followed by eight zeros. A number is written in scientific notation if it is written in the form 𝑎 multiplied by 10 to the power of 𝑏. The absolute value, or modulus, of 𝑎 must be less than 10 and greater than or equal to one. The value of 𝑏 can be a positive or negative integer.

In order to write a number in scientific notation, we firstly put a decimal point after the first nonzero digit. In this question, this is the first digit, two. As the digit after this is a zero, our value for 𝑎 will be 2.0, or just two. In order to find the value of the exponent 𝑏, we need to work out what we multiply two by to give us 200000000. We can do this using place value or by counting how many times the decimal point moves till it gets to the end of our number.

In this question, there are eight bounces, or eight place values. This means that our exponent is eight. As already mentioned, 2.0 is the same as two. So, the number 200 million in scientific notation is two multiplied by 10 to the power of eight. This is approximately the number of cows in India.

We will now look at another question using scientific notation to estimate large quantities.

The ship Titanic that sank in the Atlantic had a mass of about 47450000 kilogrammes. Which one of the following could be used as an estimate for this value? Is it A) five multiplied by 10 to the power of six? B) Five multiplied by 10 to the power of seven? C) Five multiplied by 10 to the power of eight? D) Four multiplied by 10 to the power of seven? Or E) four multiplied by 10 to the power of eight?

Our five answers are all written in scientific notation in the form 𝑎 multiplied by 10 to the power of 𝑏. We know that they’re in scientific notation as the absolute value of 𝑎 is less than 10 and greater than or equal to one. The value of 𝑏 is an integer. We need to convert 47450000 kilogrammes into scientific notation and then see which of our answers is the best estimate.

In order to write a number in scientific notation, we begin by finding the first nonzero digit. In this question, it is a four. We put a decimal point after this number and then write all the other nonzero digits. In this case, we have 4.745. This will be multiplied by 10 to some power. The exponent will be positive when dealing with large numbers and negative when dealing with small numbers. In this case, it will be a positive integer.

We need to work out how many times we would multiply 4.745 by 10 to get the answer 47450000. We can do this using place value or by counting the number of times the decimal point would move to the right. In this case, the answer is seven. 47450000 written in standard form is 4.745 multiplied by 10 to the power of seven. Looking at our five possible answers, we can immediately see that options A, C, and E are incorrect as the exponent is not seven.

In order to round our value of 𝑎, 4.745, to the nearest integer, we look at the number in the tenths column. As this is a seven, we will round up. So, our estimate becomes five multiplied by 10 to the power of seven. The correct answer is B, an estimate of 47450000 kilogrammes is five multiplied by 10 to the power of seven kilogrammes.

We’ll now look at a question converting a small number into scientific notation.

What is 0.00007 written in scientific notation?

A number is written in scientific notation if it is written in the form 𝑎 multiplied by 10 to the power of 𝑏, where the absolute value of 𝑎 must be less than 10 and greater than or equal to one and 𝑏 is a positive or negative integer. In this question, we want to write 0.00007 in scientific notation. Our first step is to find the first nonzero digit in our number. In this question, there is only one, the seven. We place a decimal point after this digit. Have there been other digits after this seven, we would’ve written these after the decimal point, for example, 7.124. In this case, there are no nonzero digits after the seven, so we have 7.0, or just seven.

We will multiply this by 10 raised to some power. Our value for 𝑏 will be negative for small numbers. This is because raising 10 to a negative power actually means we will be dividing. 10 to the power of negative four is the same as one over 10 to the power of four. So, we’re actually dividing by 10 to the power of four, which will make the number smaller.

To work out the value of 𝑏, we need to find the number of place values the numbers have moved or the number of times the decimal point has moved to the left. To get from 7.0 to 0.00007, we move our decimal point five places to the left. Therefore, our exponent is negative five. As 7.0 is the same as seven, 0.00007 written in scientific notation is seven multiplied by 10 to the power of negative five.

We will now look at a second example of writing a small number in scientific notation.

Write 0.000853 in scientific notation.

A number is written in scientific notation when it is in the form 𝑎 multiplied by 10 to the power of 𝑏. We know that the absolute value of 𝑎 must be less than 10 and greater than or equal to one. We know that 𝑏 must be an integer. It is a positive integer for large numbers and a negative integer for small numbers. In this question, we want to write the small number 0.000853 in scientific notation. We begin by finding the first nonzero digit, in this case, eight. We put a decimal point after this number and then write any further nonzero digits. In this question, our value of 𝑎 is 8.53. This is less than 10 and greater than or equal to one.

We need to multiply this by 10 raised to some power. In this case, the power or exponent will be negative as to get from 8.53 to 0.000853, we are actually dividing. Multiplying by 10 to the power of negative six is the same as dividing by 10 to the power of six. To work out the value of this exponent, we need to work out how many places our digits have moved, or how many times the decimal point has moved. In this case, it is four. Therefore, our exponent is negative four. The number 0.000853 written in scientific notation is 8.53 multiplied by 10 to the power of negative four.

Our next question involves solving a real-world problem.

The length of an object is seven millimetres. Express this length in metres, giving your answer in scientific form.

Let’s firstly recall some of our conversions of metric units of length. There are 10 millimetres in one centimetre. There are 100 centimetres in one metre. Combining these facts tells us that there are 1000 millimetres in one metre. This means that in order to convert a value from millimetres to metres, we need to divide by 1000. Seven divided by 1000 is equal to 0.007. This means that seven millimetres is the same as 0.007 metres.

This number is not in scientific form as it would need to be written 𝑎 multiplied by 10 to the power of 𝑏, where the absolute value of 𝑎 is less than 10 and greater than or equal to one and 𝑏 is an integer. 0.007 written in scientific notation is seven multiplied by 10 to the power of negative three. This is because the digits have moved three places to get from 7 to 0.007.

We could’ve worked this out directly from the calculation seven divided by 1000. 1000 is the same as 10 to the power of three, or 10 cubed. Dividing by 10 to the power of three is the same as multiplying by 10 to the power of negative three. Once again, we get the answer seven multiplied by 10 to the power of negative three. This is the length of the object in metres.

Our final question involves working out a calculation and then writing it in scientific notation.

Express three multiplied by 300 in scientific notation.

There are several ways we could approach this question. We will look at two different methods. Our first method involves multiplying three by 300 first. This gives us 900. We know that a number is written in scientific notation if it is in the form 𝑎 multiplied by 10 to the power of 𝑏, where the absolute value of 𝑎 is less than 10 and greater than or equal to one. As there is only one nonzero digit in this question, this will be the value of 𝑎.

900 written in scientific notation will be equal to nine multiplied by 10 raised to some power. 10 squared is equal to 100. And nine multiplied by 100 is equal to 900. Three multiplied by 300 expressed in scientific notation is nine multiplied by 10 squared.

An alternative method would be to rewrite three multiplied by 300 as three multiplied by three multiplied by 100. Three multiplied by three is equal to nine. 100 is equal to 10 to the power of two, or 10 squared. Once again, we get the answer nine multiplied by 10 squared.

We will now look at the key points in this video on scientific notation. A number written in scientific notation is written in the form 𝑎 multiplied by 10 to the power of 𝑏. The absolute value of 𝑎 must be less than 10 and greater than or equal to one. And 𝑏 will be a positive or negative integer. It will be positive for large numbers and negative for small numbers.

To write a number in scientific notation, we firstly find 𝑎 and then find 𝑏. 𝑎 is the number with exactly the same digits as the original number but such that the absolute value of 𝑎 is less than 10 and greater than or equal to one. 𝑏 is the number of places the decimal point needs to move from its position in 𝑎 to its position in the original number. 𝑏 will be positive for large numbers and negative for small numbers. For example, the number 237000 written in scientific notation is 2.37 multiplied by 10 to the power of five. In the same way, 0.00067 is 6.7 times 10 to the power of negative four.