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Rounding and standard form of a number

For measurements or mental calculations, we often approximate numbers to estimate the solution.

You want to estimate the surface of a square with sides of length $$212$$ metres, with no calculator.

To make it simple, you can round $$\Tred{2}12$$ to $$\Tred{2}00$$ $$$ \Tred{2}00\times \Tred{2}00 = \Tred{2}\times 100\times \Tred{2}\times 100 = 4\times 10000 = 40000.$$$ The exact solution is $$44944$$, so the leading digit is correct.

If you use $$\Tblue{21}0$$ in the calculation instead, it is more difficult but your result is closer to the exact solution, with two leading digits correct. $$$\Tblue{21}0\times \Tblue{21}0 = \Tblue{21}\times 10\times \Tblue{21}\times 10 = 441\times 100 = 44100.$$$

Approximation simplifies a number to its most important part. Estimation is the operation using the approximated numbers. The operation is simpler, but the result is not exact. The approximation error is the difference between the result and the estimate.

The error when we approximate $$212$$ by $$210$$ is $$844$$ square metres.

The number of decimal places that a number has is equal to the number of digits that appear after the decimal point.

$$1.234567$$ has six decimal places and $$1234.50$$ has two.

Numbers with many decimal places are awkward in calculations, so we can remove some of the decimal places from the right. This is called truncation.

The truncations to two decimal places of the numbers $$$ \Tblue{2.34}56,\quad \Tblue{3.45}678,\quad \Tblue{45.67}8,\quad \Tblue{-129.09}023,\quad \Tblue{12321.1}$$$ are $$$ \Tblue{2.34},\quad \Tblue{3.45},\quad \Tblue{45.67},\quad \Tblue{-129.09},\quad \Tblue{12321.1}.$$$

This display from a set of electric scales can show the mass to two decimal places.
This display from a set of electric scales can show the mass to two decimal places.

The significant figures are the digits of a given number, ignoring the leading zeros (the zeros on the left). Significant figure can be abbreviated as s.f. or sig fig.

$$0.0\Tred{1}\Tblue{230}$$ has four significant figures: 1, 2, 3 and 0. The first s.f. is $$\Tred{1}$$.

Numbers can be truncated or rounded to fewer significant figures so that they are easier to use in calculations. The significant figures that are no longer wanted are replaced by 0.

$$\Tblue{18}6$$ truncated to two significant figures is $$\Tblue{18}0$$.

$$\Tblue{18}6$$ rounded to two significant figures is $$\Tblue{19}0$$.

Here is a summary of the truncation and rounding of $$19.283$$

S.F. 1 2 3 4 5 6
Truncation $$\Tblue{1}0$$ $$\Tblue{19}$$ $$\Tblue{19.2}$$ $$\Tblue{19.28}$$ $$\Tblue{19.283}$$ $$\Tblue{19.283}$$
Rounding $$\Tblue{2}0$$ $$\Tblue{19}$$ $$\Tblue{19.3}$$ $$\Tblue{19.28}$$ $$\Tblue{19.283}$$ $$\Tblue{19.283}$$

To reduce the number of decimal places in a number, you can round it.

$$\Tblue{2.3}45$$ rounded to one decimal place is $$\Tblue{2.3}$$

$$\Tblue{2.34}5$$ rounded to two decimal places is $$\Tblue{2.3}\Torange{5}$$.

To round a number:

  • Decide how many decimal places you want to keep.
  • Look at the digit after the last digit you want to keep.
  • If this digit is $$\le 4$$, you round down and remove everything after the digit you want to keep.
  • If this digit is $$\ge 5$$, you round up by adding one to the last digit you are keeping, and then remove everything after this.

Here are a few numbers rounded to two decimal places. The red digit is the first to be removed, it determines if we round up or down. Modified digits are in orange.

$$\Tblue{2.34}\Tred{4}6$$ $$\Tblue{3.45}\Tred{6}78$$ $$\Tblue{45.89}\Tred{9}$$ $$\Tblue{-129.09}\Tred{1}23$$ $$\Tblue{12.1}$$
$$\Tblue{2.34}$$ $$\Tblue{3.4}\Torange{6}$$ $$\Tblue{45.}\Torange{90}$$ $$\Tblue{-129.09}$$ $$\Tblue{12.1}$$

Rounding causes a loss of accuracy.

If a number is rounded to a scale (unit, tenth, etc.), the original number could have differed by up to half a scale above or below the rounded value.

A number $$n$$ rounded to the nearest ten to $$\Tblue{125}0$$ must be between $$\Tblue{124}5$$ (included) and $$\Tblue{125}5$$ (excluded) $$$\Tblue{124}5\le n \lt \Tblue{125}5 $$$

The rounding error increases when we add or multiply rounded values. So, to avoid the rounding errors accumulating, only round values at the end of your calculation.

Take two numbers $$n_1$$ and $$n_2$$ rounded to the nearest $$10$$. Their possible values are covered by the ranges: $$$ \Tred{5}\le n_1\lt\Torange{15},\quad \Tred{5}\le n_2\lt\Torange{15}. $$$ We get \begin{align*} \Tred{10}=5+5\le n_1 &+n_2\lt 15+15=\Torange{30},\\ \Tred{25}=5\times5\le n_1 &\times n_2\lt15\times15=\Torange{225}. \end{align*} The biggest possible rounding error for the product ($$\Torange{225}-\Tred{25}=200$$) is bigger than the error for the sum ($$\Torange{30}-\Tred{10}=20$$).

There are three simple guidelines for rounding.

  • Two significant figures for casual measurements.

    The journey takes $$\Tblue{13}$$ minutes. The cake uses $$\Tblue{250}$$g of flour.

  • Three significant figures for accurate measurements. These might be scientific measurements, or measurements where accuracy is important, such as building work.

    The moon orbits the Earth in $$\Tgreen{27.5}$$ days. Its axis is tilted at $$\Tgreen{6.58}$$ degrees from the vertical.

  • More than three significant figures for very accurate scientific measurements.

    The speed of light is $$\Torange{299,\!792,\!458}$$ metres per second.

The journey from London to Paris usually takes 2 hours and 20 minutes by train.
The journey from London to Paris usually takes 2 hours and 20 minutes by train.

For very small or very big quantities, we use metric prefixes depending on the factors of tens that apply to the number.

A millisecond is a thousandth of a second $$(1\textrm{ ms} = 10^{-3}\us)$$. A bee's wing flaps every $$5\textrm{ ms}$$.

A gigametre is one billion meters $$(1\textrm{ Gm} = 10^{9}\um)$$. The diameter of the Sun is $$1.4\textrm{ Gm}$$.

Name Trillion Billion Million Thousand Hundred Ten
$$10^n$$ $$10^{\Tred{12}}$$ $$10^\Tred{9}$$ $$10^\Tred{6}$$ $$10^\Tred{3}$$ $$10^\Tred{2}$$ $$10^\Tred{1}$$
Prefix Tera Giga Mega Kilo Hecto Deca
Symbol T G M k h da
$$10^{-n}$$ $$10^{\Torange{-12}}$$ $$10^{\Torange{-9}}$$ $$10^{\Torange{-6}}$$ $$10^{\Torange{-3}}$$ $$10^{\Torange{-2}}$$ $$10^{\Torange{-1}}$$
Prefix Pico Nano Micro Milli Centi Deci
Symbol p n $$\mu$$ m c d

Small or big numbers are generally written in standard form or scientific notation. This makes the numbers easier to write, shows the scale of the number, and makes it easy to compare quantities.

A positive number is in standard form if it is written as $$$ \Tblue{A} \times 10^\Tred{n}.$$$ The number $$\Tblue{A}$$ must be between 1 (included) and 10 (excluded) in absolute value. It is sometimes called the mantissa. The number $$\Tred{n}$$ is an integer, called the exponent.

$$7600$$ is written in standard form $$\Tblue{7.6}\times 10^\Tred{3}$$. The mantissa is $$\Tblue{7.6}$$ and the exponent is $$\Tred{3}$$. The scientific notation for $$-0.02$$ is $$\Tblue{-2}\cdot 10^{\Tred{-2}}$$

Some examples:

\begin{align*} 123.4 & = \Tblue{1.234}\times 10^\Tred{2},\quad-32,\!124 = \Tblue{-3.2124}\times 10^{\Tred{5}},\\ &0.32 = \Tblue{3.2}\times 10^{\Tred{-1}},\quad 0.032 = \Tblue{3.2}\times 10^{\Tred{-2}}. \end{align*}

The exponent gives the position of the first significant figure.

$$100 = 10^\Tred{2}$$ has $$\Tred{2}+1 = 3$$ digits left from the decimal point.

$$0.01 = 10^{\Tred{-2}}$$ has $$\Tred{2}-1 = 1$$ zero right from the decimal point.

Saturn is 116 000 000 m in diameter. E. coli cells are 0.000 002 m long. These numbers are easier to handle using standard form.
Saturn is 116 000 000 m in diameter. E. coli cells are 0.000 002 m long. These numbers are easier to handle using standard form.

To add or subtract numbers in standard form, convert them into decimal notation, do the operation and convert them back. You can also isolate the power of ten with the lowest exponent to make the calculation simpler.

\begin{align*} \Tblue{3.6} \times 10^\Tred{3} + \Tblue{7.1} \times 10^\Tred{2}= 3600 + 710 = 4310 & = \Tblue{4.31} \times 10^\Tred{3}\\ \qquad= \Tblue{36}\times 10^\Tred{2} + \Tblue{7.1}\times 10^\Tred{2} = \Tblue{43.1}\times 10^\Tred{2} & = \Tblue{4.31}\times 10^\Tred{3} \end{align*}

To multiply or divide numbers in standard form, use the laws for indices. Indices are added for multiplication and subtracted for division.

\begin{align*} (\Tblue{6.2} \times 10^\Tred{5}) \times (\Tblue{3.1} \times 10^\Tred{2}) &= (\Tblue{6.2\times 3.1}) \times 10^{\Tred{5+2}}\\ &= \Tblue{19.22} \times 10^\Tred{7}\\ &= \Tblue{1.922} \times 10^\Tred{8}\\ (\Tblue{6.2} \times 10^\Tred{5}) \div (\Tblue{3.1} \times 10^\Tred{2}) &= (\Tblue{6.2 \div 3.1}) \times 10^{\Tred{5-2}} \\ &= \Tblue{2} \times 10^\Tred{3} \end{align*}

Large numbers are often given in standard form. How far is Saturn from Earth?
Large numbers are often given in standard form. How far is Saturn from Earth?