Scientific notation standard form is an effective and often convenient method of writing very big or very small numbers in a shorter way.

We can see learn how converting numbers to scientific notation works by first looking at an example.

### Converting Numbers to Scientific Notation

Example:

Consider the fairly large number seven thousand.

**7’000**

This number can be written in scientific notation form.

If we think about what makes seven thousand as a number, it is seven thousands.

So effectively one thousand seven times, a sort of multiplication.

**7**×

**1’000**

The one thousand can also be written as 10

^{3}. ( As 10

^{3}= 10 × 10 × 10 = 1’000 )

So with seven thousand we can write:

**7**×

**10**

^{3}.

Which is the scientific notation standard form of seven thousand.

This is the case with other number also, leading us to the general form of scientific notation.

### Scientific Notation General Case:

Numbers is scientific notation form are written as:N × 10

^{r}

Where

*r*is an integer, and N is number between 1 and 10 which can be a whole number or a decimal number.

The scientific notation standard form examples below have been split into two sections, one for writing larger numbers and one for writing smaller numbers.

As the approach is slightly different depending on which type if number we’re dealing with.

## Scientific Notation Examples

### Larger Numbers

*(1.1)*Write

**5’000’000**in scientific notation.

*Solution*This is the large number five million.

Which we want to change to the form N × 10

^{r}, adhering to the conditions specified above.

An effective method is to write the number out, and place a decimal point on the far right side of the number.

Then move the point to the left until only the first digit is on the left of the point.

5’000’000

**.**Point moving 6 places to the left. 5

**.**000’000

This 5 will be our N in scientific notation.

While our

*r*will be 6, the number of places we moved the decimal point.

So 5’000’000 in scientific notation is

**5**×

**10**

^{6}.

*(1.2)*Write

**340’000**in scientific notation.

*Solution*Here we have the number three hundred and forty thousand.

The same approach used in example (1.1) can be used here.

340’000

**.**Point moves 5 places to the left. 3

**.**40’000

There is an extra digit to the left of the point here, but it’s nothing to worry about.

It’s just the case that 3

**.**4 will be N in the scientific notation, and like before,

*r*will be the number of places the point moved, 5.

**340’000**=

**3.4**×

**10**

^{5}

### Smaller Numbers

*(2.1)*Write

**0.002**in scientific notation.

*Solution*Here we have a small number to convert.

The approach is a bit different than with larger numbers as seen above.

We move the decimal point in the number along to the right, until there is just the one non zero digit on the left of the point.

0

**.**002 Point moves 3 places to the right. 0002

**.**

Now from here, 2 will be N, and

*r*will be –3, according to the places the point moved.

**0.002**=

**2**×

**10**

^{-3}

To understand why the exponent is now negative in nature, we can construct 0.002 as a sum.

**0.002**=

^{\bf{\frac{2}{1000}}}=

**2**×

^{\bf{\frac{1}{1000}}}(

^{{\frac{1}{1000}}}=

^{{\frac{1}{10^3}}}= 10^{{\text{-}}3} )

*(2.2)*Write

**0.0000546**in scientific notation.

*Solution*Again we move the decimal point in the number along to the right, until there is one non zero digit on the left of the point.

0

**.**0000546 Point moves 5 places to the right. 000005

**.**46

Now, 5

**.**46 will be N, and

*r*will be –5, according to the places the point moved.

**0.0000546**=

**5.46**×

**10**

^{-5}

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