Approximations

There are two ways to represent our answers in math, the exact value and approximate value. The exact value (the true value) preserves the integrity of the number, meaning that we do not round any digits e.g. Pi ($\pi$). By rounding, we are throwing away information (the digits lost by rounding).

The approximate value, on the other hand, is close to the true value but only to a certain extent (such as a specific number of decimal places, etc.) e.g. $3.14$ is Pi written to $2$ decimal places. The value $3.14$ is thus referred to as an approximation of Pi ($\pi$). That is, it is close to Pi but not exactly Pi.

To approximate is to simply use a value close to the true value. There are several ways in which we perform approximations in math and science:

• Decimal places
• Significant figures
• Scientific notation a.k.a. standard form

Significant figures

We rewrite the number using the following rules:

• Non-zero digits are significant
• Leading zeros are not significant
• Trailing zeros are significant

In short, the first significant figure cannot be zero.

Challenge

Write the following numbers to the specified number of significant figures:

• $4570$ to $2$ s.f.
• $0.007311$ to $1$ s.f.
• $0.9013$ to $3$ s.f.
• $104.200$ to $3$ s.f.

Challenge

State the number of significant figures in the following numbers:

• $0.08560$
• $600,000$
• $203.50$

Scientific notation/standard form

Sometimes we want to represent very large or very small numbers e.g. $4100000000$, $0.000000003$, etc. In the process of writing these numbers, we have to write many, many zeros. Standard form enables us to avoid having to write all of these zeroes. When writing a number in standard form, the format is

$$ \begin{equation}\begin{aligned} a\times 10^b\\ \end{aligned}\end{equation} $$

Where $a$ is a non-zero single digit number and $b$ is an integer. For example, the number $4100000000$ will be $$ \begin{equation}\begin{aligned} 4.1\times 10^9\\ \end{aligned}\end{equation} $$

How to rewrite in standard form

Consider writing the number $345$ in standard form. We write the original number as the product of itself and $10^0$:

$$ \begin{equation}\begin{aligned} 345\times 10^0\\ \end{aligned}\end{equation} $$

We then shift the decimal point is in such a way that there is only one non-zero single digit before the decimal point, in this case, $3$:

$$ \begin{equation}\begin{aligned} 3.45\times 10^{0+2}\\ \end{aligned}\end{equation} $$

Because we shifted the decimal point $2$ places to the left, we increase the power of $10$ by $2$. Therefore the number $345$ written in standard form is $3.45\times 10^{2}$.

Example 2

Write the number $0.0037$ in standard form. First we write it as itself times $10^0$: $$ \begin{equation}\begin{aligned} 0.0037\times 10^0\\ \end{aligned}\end{equation} $$

Then we shift the decimal point $3$ places to the right, decreasing the power of $10$ by $3$: $$ \begin{equation}\begin{aligned} 3.7\times 10^{0-3}\\ \end{aligned}\end{equation} $$

Thus the number $0.0037$ written in standard form is $3.7\times 10^{-3}$.

Challenge

Write the following numbers in standard form:

• $53,000$
• $0.0007205$
• $901,500,000$
• $0.000034$