# Standard form

This is such a very huge number to read or write. It would be incredibly hard to use in any maths calculation and you wouldn’t be able to enter such a number in a calculator. It is also very hard to remember! You would have to find some easy way to write it. This is where a standard form expression is useful. This number can be written as;

^{24}

As you can see the number is very easy to remember and it would be easy to use in any maths calculation since it has a few digits.

Similarly with this number it is just too small to work with, it has too many numbers of 0s and looks to busy to remember. This number can be written as;

^{-27}

The following table offers an overview of the kind of numbers that can be rewritten in standard form.

Standard form | …means | Actual value |
---|---|---|

2 x 10^{3} | 2 x 10 x 10 x 10 | 2000 |

2 x 10^{2} | 2 x 10 x 10 | 200 |

2 x 10^{1} | 2 x 10 | 20 |

2 x 10^{0} | 2 x 1 | 2 |

2 x 10^{-1} | 2 x 0.1 | 0.2 |

2 x 10^{-2} | 2 x 0.1 x 0.1 | 0.02 |

2 x 10^{-3} | 2 x 0.1 x 0.1 x 0.1 | 0.002 |

## Writing intergers in standard form

- 40,000
- 450,000

- 40,000 = 4 × 10
^{4} - 450,000 = 4.5 × 10
^{5}

We shall write 40,000 in standard form first. We follow a few basic steps.

Step 1: Find the highest multiple between 1 and 10 but less than 10Find the highest possible multiple between 1 and 10 but less than 10. It can be a decimal number. In this example the highest multiple is a 4. Or you choose the first integer of the number 40,000. In this case 4.

40,000 = 4 × 10,000Step 2:Find the number of 10 powersWe have to find the number 10 powers that can be multiplied with 4 to get 40,0000. A number that can be multiplied with 4 to get 40,000 is 1000. Note 10,000 can be coverted into an indice which is much easier to remember.

10,000 = 10^{4}Step 3:Rewrite in standard form.40,000 can be written in standard form by simply multiplying 4 by the integer.

40,000 = 4 × 10^{4}When writing numbers into standard form you must remember this very important rule;

The first number must always be a number between 1 and less than 10 but not 10.Next we shall write 450,000 in standard form. Note there is a difference between this number and the previous number. Again we follow the same steps.

Step 1: Find the highest multiple between 1 and 10 but less than 10Find the highest possible multiple between 1 and 10 but less than 10. This number can be a decimal number. Or we could pick the first non zero integers and create the smallest decimal number possible. You do this by putting the decimal number after the first integer. In this case the number is 4.5

450,000 = 4.5 × 100,000Step 2: Find the number of 10 powers.We have to find the number of 10 powers that can be multiplied with 4.5 to get 450,000. In this case 100,000

100,000 = 10^{5}Step 3: Rewrite in standard formGiven what we have found out about the number we can rewrite it in standard form.

450,000 = 4.5 × 10^{5}

## Writing decimals in standard form

^{-7}

To write this number in standard form we follow a few basic steps. Just like we did before.

Find the highest possible multiple between 1 and 10 but less than 10. In this case the number is 6.

We must find the number of powers that when multiplied with should provide 0.0000006. The number of powers is 0.0000001. When working with decimals you find this by counting the number of digits after the first decimal. This number of digits will represent a negative number power of 10. In this case there is 7 digits after the decimal point.

^{-7}

We have found 10^{-7}. We can use this as a factor for the highest multiple to rewrite 0.0000006 in standard form.

^{-7}

^{-4}

We follow the same steps as we did in the previous example.

Find the highest possible multiple between 1 and 10 but less than 10. In this case the number is 4.34. You have to remember that the first number is always less that 10 (positive)

We must find the number of powers that when multiplied with should provide 0.0001.

^{-4}

Using what we know about the number we can rewrite it as;

^{-4}

You must always remember when working with standard forms the first number is always between 1 and less than 10

For example; 13 x 10^{4} is not in standard form, neither is 0.13 x 10^{-3}. The correct standard form is; **1.3 x 10 ^{4} ** or

**1.3 x 10**respectively. Notice the first number is between 1 and less than 10. Writing numbers greater that 10 is a very common mistake to make.

^{-3}