Scientific Notation

Background

Sometimes, we want to get a sense of a number, an approximation, that tells just how big it is. Scientific Notation can be used to do that.

Question

Express these in scientific notation: 3000, 0.0005, 45245, 0.00345

Calculate 3000 x 0.0005 

Answer

3000 = 3 x 10^3, 0.0005 = 5 x 10^(-4), 45245 = 4.5245 x 10^4, 0.00345 = 3.45 x 10^(-3)

3000 x 0.0005 = 1.5 

Analysis

When we're looking at numbers, particularly very big and very small numbers, they can be hard to manage. For instance, what's the difference between 1,000,000,000,000 and 1,000,000,000,000,000 - notice that we have to count zeros and apply words to it - it's difficult to deal with. What would be great is if we could see in an easy way the size of a number. And we can.

We can take a number and express it so that there is an integer between 9 and -9, with the remaining digits behind the decimal point. The size of the number can be seen as a power of 10.

For instance, looking at the number 3000, we can express it as:

3 x 1000

and we can rewrite 1000 in terms of an exponential:

3 x 10^3

We can do the same with a very small number, say 0.0005. We can express it as:

5 x 0.0001

and we can rewrite 0.0001 in terms of an exponential:

5 x 10^(-4)

This is less useful if we have a lot more digits or is smaller. Look at 45245:

45245 = 4.5245 x 10,000 = 4.5245 x 10^4

Or this number: 0.00345

0.00345 = 3.45 x 0.001 = 3.45 x 10^(-3)

We can use multiplication rules to multiply scientific notation numbers in an easy manner. We can take, say 3000 x 0.0005, and through conversion to scientific notation we can easily multiply these numbers:

3000 x 0.0005 = 3 x 10^3 x 5 x 10^(-4) = 15 x 10^(-1) = 1.5 x 10 x 10^(-1) = 1.5

Vocabulary used:

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Where might you have come from?

Fact-orials Index

Numbers:

• Real Numbers
• Expressing Values and Bases

Associated Operations:

• Estimation and Rounding

Where might we go?