Lesson 51: Scientific Notation for Large Numbers

New Concept

This course builds a strong mathematical foundation, mastering arithmetic and introducing key concepts like fractions, decimals, and percents that describe the world around us.

What’s next

We begin by exploring powerful ways to handle very large numbers using a tool called scientific notation. You’ll see how it simplifies complex calculations.

Property

Scientific notation is a way of expressing numbers as a product of a decimal number and a power of 10. A number in scientific notation has the form $c \times 10^n$, where the factor $c$ is greater than or equal to 1 but less than 10.

Examples

• The number $540,000,000$ becomes much shorter as $5.4 \times 10^8$.
• To read $8.12 \times 10^7$, you just move the decimal 7 places right to get $81,200,000$.
• A light-year, which is $9,461,000,000,000$ km, is written simply as $9.461 \times 10^{12}$ km.

Explanation

This is a cool way to write gigantic or tiny numbers without a ton of zeros. Think of it like a secret code! You write down the main digits, then use a power of 10 to tell everyone how many places to move the decimal point. It makes writing the distance to a star way easier!

Property

Read the number $9.461 \times 10^{12}$ as 'Nine point four six one times ten to the twelfth.'

Examples

• The number $4.25 \times 10^5$ is read as 'Four point two five times ten to the fifth.'
• The number $7.0 \times 10^9$ is read as 'Seven point zero times ten to the ninth.'
• The number $2.987 \times 10^3$ is read as 'Two point nine eight seven times ten to the third.'

Explanation

Don't let the symbols trick you! Just read the decimal part as you normally would, then say 'times ten to the,' and finish with the exponent's power. It sounds super scientific, but it’s actually just a straightforward way to say a number. You are just giving verbal instructions on how the number is built.

Property

To write a number in scientific notation, place the decimal point to the right of the first nonzero digit. Then, count the number of places the decimal moved from its original position to determine the exponent for the power of 10.

Examples

• For $81,000,000$: place the decimal after 8 to get $8.1$. The decimal hopped 7 places, so it's $8.1 \times 10^7$.
• For $602,500,000$: place the decimal after 6 to get $6.025$. The decimal hopped 8 places, so it's $6.025 \times 10^8$.
• A football field is $110,000$ mm, which becomes $1.1 \times 10^5$ mm after the decimal hops 5 places.

Explanation

Think of it as pinning the decimal right after the first important digit. Then, count how many hops the decimal made from where it started to its new home. That number of hops becomes your exponent for the power of 10. This trick lets you neatly package huge numbers without writing all those trailing zeros.