Multiplying Decimal Numbers

Procedure

1. Ignore the decimal points - they are not important until you reach step three.

2. Multiply the two figures.

3. Count the number of decimal places of both coefficients and add them up. In this case, each figure has one decimal place. Take the result of your multiplication and place a decimal mark in a way that the result receives as many decimal places as both coefficients combined ( 2 in this example.)

Remarks

There are a few ways how to improve the notation when multiplying decimal numbers.

1. Zeros at the end of the solution

$2.5\cdot 1.2\to 25\cdot 122.5\backslash cdot1.2\backslash rightarrow25\backslash cdot12$

$\begin{array}{c}\underset{‾}{25\cdot 12}\\ \phantom{000,}50\\ \underset{‾}{\phantom{20,}250}\\ \phantom{20,}300\end{array}\backslash def\backslash arraystretch\{1.25\}\; \backslash begin\{array\}\{l\}\backslash underline\{25\backslash cdot12\}\backslash \backslash \backslash hphantom\{000,\}50\backslash \backslash \backslash underline\{\backslash hphantom\{20,\}250\}\backslash \backslash \backslash hphantom\{20,\}300\backslash end\{array\}$

$\phantom{20}3.00=3\backslash hphantom\{20\}3.00\; =\; 3$

A zero at the end of the result must be taken into account when counting. After setting the point, however, it can then be omitted as usual.

2. Simpler notation

In order to write a bit less, one can also leave the points in the top line and finally add them to the solution.

$2.1\cdot 3.22.1\backslash cdot3.2$

$\to 21\cdot 32\backslash rightarrow\; 21\backslash cdot32$

$\begin{array}{c}\underset{‾}{21\cdot 32}\\ \phantom{000,}42\\ \underset{‾}{\phantom{20,}630}\\ \phantom{20,}672\end{array}\backslash def\backslash arraystretch\{1.25\}\; \backslash begin\{array\}\{l\}\backslash underline\{21\backslash cdot32\}\backslash \backslash \backslash hphantom\{000,\}42\backslash \backslash \backslash underline\{\backslash hphantom\{20,\}630\}\backslash \backslash \backslash hphantom\{20,\}672\backslash end\{array\}$

$\phantom{20}6.72\backslash hphantom\{20\}6.72$

This turns the long exercise solution into the following shorter one:

$\begin{array}{c}\underset{‾}{2.1\cdot 3.2}\\ \phantom{-2000}42\\ \underset{‾}{\phantom{-200}630}\\ \phantom{200,}6.72\end{array}\backslash def\backslash arraystretch\{1.25\}\; \backslash begin\{array\}\{l\}\backslash underline\{2.1\backslash cdot3.2\}\backslash \backslash \backslash hphantom\{-2000\}42\backslash \backslash \backslash underline\{\backslash hphantom\{-200\}630\}\backslash \backslash \backslash hphantom\{200,\}6.72\backslash end\{array\}$

Rules of computation

While multiplying decimal numbers, watch out for the following special cases:

• The usual laws of multiplication also apply to decimal numbers. This means that, as with the multiplication of integer numbers, the commutative law, the associative law and, with addition, the distributive law apply.

• An easy special case is the multiplication with $\mathrm{10,100,1000},\dots 10\{,\}100\{,\}1000,\dots$. The decimal point is simply shifted to the right by as many places as there are zeros in the coefficient. Example: $3.4\cdot 100=34\cdot 10=3403.4\; \backslash cdot\; 100=34\backslash cdot\; 10=340$

• Similar to the case detailed above, multiplying with $0.1;0.1;0.01;\dots 0.1;\; 0.1;\; 0.01\; ;\dots$ shifts the decimal point to the left by one place per zero in the coefficient. Do not forget the zero in front of the decimal mark! Example: $3.4\cdot 0.01=0.34\cdot 0.1=0.343.4\backslash cdot\; 0.01=0.34\backslash cdot\; 0.1=0.34$