Scientific notation is a way of representing very large or very small numbers, without needing to write lots of zeros. Consider the mass of the sun, which is approximately 1,988,000,000,000,000,000,000,000,000,000 \text{ kg}. That's a very large number. How do scientists deal with numbers so large?

Exploration

This applet shows how a decimal number is written in scientific notation.

Click the "Generate a random number" button to explore different numbers.
Check the box to see what each number looks like written in scientific notation.

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Explore 10 random numbers and how the power of ten relates to the number in the tens position.
What types of numbers have positive or negative powers of ten?
Describe the relationship between the place value of the digits and the number in scientific notation.
Can you come up with a rule for taking a very large number in scientific notation? What about a very small number?

Going back to the example about the mass of the sun:

The sun has a mass of approximately 1.988\times 10^{30} \text{ kg} , which is much easier to write than 1,988,000,000,000,000,000,000,000,000,000 kg.
The mass of an atom of Uranium (one of the heaviest elements) is approximately 3.95\times 10^{-22} g. That is 0.000\,000\,000\,000\,000\,000\,000\,395 g.

In scientific notation, numbers are written in the form:

\displaystyle a \times 10^{n}

\bm{a}

decimal number greater than or equal to 1 but less than 10

\bm{n}

positive or negative integer

A negative exponent indicates how many factors of ten smaller than a the value is.
A positive exponent indicates how many factors of ten larger than a the value is.

We can follow these steps in writing numbers in scientific notation.

Move the decimal point to the left or right so that it is right after the first non-zero digit (from 1to 9).
Where's the decimal point in 2,680,000? Because it's a whole number, the decimal point is understood to be at the end of the number: 2,680,000.
The first non-zero number is 2. If we move the decimal point 6 places from the end of the number to the right of the 2, we will get 2.68. We don’t need the extra zeroes. The number 2.68 is between 1 and 10 as we wanted.
Multiply by 10 to the power of the number of places the decimal moved.
We moved 6 places to the left so we have 10^{6}.

Standard form	Product form	Scientific notation
2,680,000	2.68\times 1,000,000	2.68\times 10^{6}

Remember, we're not actually "moving" the decimal point. We're adjusting the place value. Then to balance that out, we have to multiply by the correct power of 10.

Examples

Example 1

What value should go in the space to make the expression on the right in scientific notation form? 300 = ⬚ \times 10^{2}

Worked Solution

Create a strategy

Write the expression without scientific notation.

Apply the idea

10^{2} is equivalent to 10\times 10 or 100.

300 = 3 \times 10 \times 10

300 = 3 \times 10^{2}

The value that should go in the space is 3.

Reflect and check

This means that 300 in scientific notation is 3\times 10^{2}.

Note that the definition of scientific notation form is a \times 10^{n}, where a is greater than or equal to 1 and less than 10. This is why we are putting 3 into the blank space, and not 30 or 300.

Example 2

Express 0.07 in scientific notation.

Worked Solution

Create a strategy

Since the number is less than 1, we will move the decimal point to the right so that we have the decimal point right after the first non-zero single digit.

Apply the idea

We'll move the decimal point of 0.07 two places to the right so we'll have 7.0 or simply 7.

Moving to the right means a negative power of 10.

The number 0.07 expressed in scientific notation is 7\times 10^{-2}.

Example 3

Express 3.66\times 10^{-6} as a decimal number.

Worked Solution

Create a strategy

Since the power is negative, the answer will be a smaller decimal.

Apply the idea

We should move the decimal point 6 places to the left, and fill the spaces with zeros.

\displaystyle 3.66 \times 10^{-6}

\displaystyle =

\displaystyle 0.000\, 003\,66

Move the decimal point

Example 4

Scientists recorded the diameter of Mercury as 4.879 \times 10^{3}\text{ km}. Write the diameter of mercury in standard form.

Worked Solution

Create a strategy

Find the value of the power of ten then multiply it by the constant term to get the number in standard form.

Apply the idea

\displaystyle \text{Diameter of mercury}	\displaystyle =	\displaystyle 4.879\times 10^{3}	Given
	\displaystyle =	\displaystyle 4.879 \times 1000	Evaluate the power
	\displaystyle =	\displaystyle 4879\text{ km}	Evaluate the multipication

Idea summary

A number is written in scientific notation if it has the form a \times 10^{n} where a is greater than or equal to 1 and less than 10, and n is an integer.

Standard form	Product form	Scientific notation
2,680,000	2.68\times 1,000,000	2.68\times 10^{6}
0.000794	7.94\times \dfrac{1}{10,000}	7.94\times 10^{-4}

Outcomes

7.NS.1

The student will investigate and describe the concept of exponents for powers of ten and compare and order numbers greater than zero written in scientific notation.

7.NS.1c

Convert between numbers greater than 0 written in scientific notation and decimals.*

1.06 Scientific notation

Ideas

Scientific notation

Exploration

Examples

Example 1

Example 2

Example 3

Example 4

Outcomes

7.NS.1

7.NS.1c

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