# Standard form

**Powers of ten**

The decimal number system is based on **powers of ten**. This can be written as powers of ten using an **index notation**.

10 = 10¹

100 = 10 x 10 = 10²

1000 = 10 x 10 x 10 = 10³

10 000 = 10 x 10 x 10 x 10 = 10⁴

100 000 = 10 x 10 x 10 x 10 x 10 = 10⁵

1 000 000 = 10 x 10 x 10 x 10 x 10 x 10 = 10⁶…

Points to note:

- The index tells us how many 0s will follow the 1, which is only true for positive integers of 10.
- Any number raised to the power of 0 is 1, so

Decimals can be written using negative powers of ten

** Very large numbers**

15 to the power of 10 multiplied by 2 means = 2 followed by 15 zeros or:

2 x 10¹⁵ = 2 000 000 000 000 000

The opposite can also be done for a number is a large amounts of decimal places.

For example:

–12 to the power of 10 multiplied by 1.5 = 15 is **divided** by 1 followed by 12 zeros (which includes the zero before the decimal point) .

1.5 x 10¯¹² = 0.000000000002

**Standard form**

In the examples above, 2 x 10¹⁵ adn 1.5 x 10¯¹² are numbers written in **standard form**.

Numbers written in standard form have two parts:

- A number between 1 and 10
- A power of 10

This is also known as a **standard index form** or **scientific notation**. Note also that any number can be written using a standard form but this is usually used for very large or very small numbers.

**Ordering numbers in standard form**

Write these numbers in order from smallest to largest”

5.3 x 10¯⁴, 6.8 x 10¯⁵, 4.7 x 10¯³, 1.5 x 10¯⁴

To order numbers that are written in standard form, follow the following steps:

1. Start by comparing the powers of 10.

2. is smaller than and vice versa

3. If the powers of 10 are the same, compare the number parts

There the solution is:

6.8 x 10¯⁵, 1.5 x 10¯⁴, 5.3 x 10¯⁴, 4.7 x 10¯³

**Calculations involving standard form**

To calculate numbers involving the standard form, the following steps should be taken:

For multiplication:

1. Multiply the number parts together and then the powers of ten together. (indices are added when we multiply)

2. Ensure the number part is in standard form, if it is more than 10, then add an additional power of 10.

For division

1. Divide the number parts first, then the powers of 10. (indices are subtracted when we divide).

2. Ensure the number part is in standard form, if there is an additional negative power of 10, then add an additional negative power of 10.