# Standard Form

## Standard Form Revision

**Standard Form**

Sometimes in **b****iology**, **chemistry **and **physics**, we have the deal with very large and very small numbers. **Standard form** is a way of expressing these numbers, and it saves you from having write out the numbers in their entirety. You will need to know how to convert normal numbers into **standard form **and vice versa.

**What is Standard Form?**

**Standard form** is always written as number between 1 and 10, multiplied by 10 to the power of a whole number. Here are some examples:

5.4 \times 10^7

6.2 \times 10^{-3}

1.1 \times 10^{23}

6.63 \times 10^{-34}

The **positive powers** indicate **bigger numbers **and the **negative powers** indicate **smaller numbers**.

**Writing Numbers in Standard Form**

It is useful to write both very small and very large numbers in **standard form**.

**Example 1:** Write 15 100 in **standard form**.

We need to write this as a number between 1 and 10. We know this must be 1.5. We can then count how many times the decimal place moves over:

The decimal place has moved over 4 times, from \boldsymbol{15100} to \boldsymbol{1.51}. The number of times the decimal moves over is the power. The power will be positive because the number is a big one (also because the decimal place has moved to the right). So the power will be positive 4. Therefore we can write 15100 as 1.51 \times 10^4.

**Example 2:** Write 0.000071 in **standard form**.

We use the same method as the previous example. The number between 1 and 10 will be 7.1. We now count how many times we move the decimal place to get to 7.1:

The power is negative because the number is a small one (also because the decimal place has moved to the right). Therefore, 0.000071 in **standard form** is 7.1 \times 10^{-5}.

**Converting from Standard Form**

Converting from **standard form** is simple once you know how to write numbers in **standard form**. You simply count either forwards or back, moving the decimal place by the number of the power.

**Example 1:** Convert 3.15 \times 10^{7} from standard form.

The power is a positive 7 so need to move the decimal place 7 spaces up the number (to the right), and add zeros in the missing spaces:

Therefore 3.15 \times 10^7 converted from** standard form** is 31500000.

**Example 2:** Convert 6.78 \times 10^{-4} from** standard form**.

The power is a negative 4 so need to move the decimal place 4 spaces down the number (to the left) and add zeros in the missing spaces:

Therefore 6.78 \times 10^{-4} converted from **standard form** is 0.000678.

## Standard Form Example Questions

**Question 1:** Write 52567 in standard form.

**[1 mark]**

\boldsymbol{5.2567 \times 10^4}

**Question 2:** Write 0.009061 in standard form.

**[1 mark]**

\boldsymbol{9.061 \times 10^{-3}}

**Question 3:** Determine which of the following numbers is larger.

5.36 \times 10^{-3}

7.82 \times 10^{-2}

**[1 mark]**

Converting each from standard form:

5.36 \times 10^{-3} = 0.00536

7.82 \times 10^{-2} = 0.0782

0.0782 > 0.00536 therefore 7.82 \times 10^{-2} is the larger number.