# Standard Form

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## Standard Form

Sometimes in biology, 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.

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## 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 $7.1 \times 10^{-5}$

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## 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 $0.000678$.

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@mmerevise

## Standard Form Example Questions

$\boldsymbol{5.2567 \times 10^4}$

Gold Standard Education

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

Gold Standard Education

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.