Approximation and Errors

Hong Kong

Stage 1 - Stage 3

Lesson

Scientific notation is a way of writing very big or very small numbers in a nice, compact way (because we all know mathematicians like to shorten everything). Funnily enough, scientific notation is frequently used in science. For example, the sun has a mass of $1.988\times10^{30}$1.988×1030kg which is much easier to write than $1988000000000000000000000000000$1988000000000000000000000000000kg.

Now let's look at an example of a calculation with scientific notation.

Suppose you had to multiply $1000000$1000000 by $1000000000$1000000000. Although the multiplication process is quite straightforward (you just add up the total number of zeros), it is very easy to miscount the number of zeros in each number.

Likewise, multiplying $0.000001$0.000001 by $0.000000001$0.000000001 is straightfoward (you just add up the total number of zeros after the decimal points), but again it would be very easy to make an mistake.

This is where scientific notation comes in handy.

For example, the first product above can be rewritten as $10^6\times10^9$106×109 and the computation can be carried out by simply summing the exponents using the multiplication law with little chance of making an error, so our answer would be $10^{15}$1015.

Likewise, the second product above can be rewritten as $10^{-6}\times10^{-9}$10−6×10−9. Again, we can easily compute this by adding the powers using the multiplication law, which would give us an answer of $10^{-15}$10−15.

In scientific notation, numbers are written in the form $a\times10^b$`a`×10`b`, where $a$`a` is a number between $1$1 and $10$10 and $b$`b` is any integer (positive or negative) that is expressed as an index of $10$10. If you need a refresher on how to multiply or divide by factors of 10, click here.

Remember

- A
**negative**power indicates how many times**smaller**the $a$`a`value will be. - A
**positive**power indicates how many times**larger**the $a$`a`value will be.

Express $63300$63300 in scientific notation.

**Think:** We need to express the first part of scientific notation as a number between $1$1 and $10$10 and then work out the index of $10$10 that it is raised by.

**Do:**

To express this value is a value between $1$1 and $10$10, we can write it as $6.33$6.33.

$63300$63300 is $1000$1000 or $10^4$104 time bigger than $6.33$6.33.

So in scientific notation, we would write this as $6.33\times10^4$6.33×104.

Evaluate $15^{-4}$15−4 correct to 3 significant figures. Give your answer in scientific notation.

If we round to $1$1 significant figure, sound travels at a speed of approximately $0.3$0.3 kilometres per second, while light travels at a speed of approximately $300000$300000 kilometres per second.

Express the speed of sound in kilometres per second in scientific notation.

Express the speed of light in kilometres per second in scientific notation.

How many times faster does light travel than sound?