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Mathematics LibreTexts

2.2: The Commutative Property of Addition and Multiplication

  • Page ID
    13968
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    Figure 2.2.1.png

    Figure 2.2.1

    \(2\) commuted (traveled) from position 1 to position 2

    and

    \(3\) commuted (traveled) from position 2 to position 1.

    Note

    A commuter is a traveler. Do not say “communitive" property.

    Figure 2.2.1 illustrates the commutative property of addition.

    In general

    \[\boxed{\Large a+b=b+a}\]

    where \(a\) and \(b\) are any real numbers (like \(-6.4\), \(\displaystyle \frac{2}{7}\), \(\pi\)). Any real number can hide in the \(a\)-box or the \(b\)-box.

    Example \(\PageIndex{1}\): Is subtraction commutative?

    Is \(3-1=1-3\) a true statement?

    Solution

    No, because \(3-1-2\) and \(1-3=-2\). If we find one counterexample, one example that shows that subtraction is not commutative, the general property (using \(a\) and \(b\)) does not exist.

    Figure 2.2.2.png

    Figure 2.2.2

    \(2\) commuted (traveled) from position 1 to position 2

    and

    \(3\) commuted (traveled) from position 2 to position 1.

    Figure 2.2.2 illustrates the commutative property of multiplication.

    In general

    \[\boxed{\Large ab=ba}\]

    where \(a\) and \(b\) are any real numbers (like \(-6.4\), \(\displaystyle \frac{2}{7}\), \(\pi\)). Any real number can hide in the \(a\)-box or the \(b\)-box.

    Note that \(ab\) means \(a\) times \(b\).

    Example \(\PageIndex{2}\): Is division commutative?

    Is \(4\div 2=2\div 4\) a true statement?

    Solution

    No, because \(4\div 2=2\) and \(2\div 4=\displaystyle \frac{1}{2}=0.5\). If we find one counterexample, one example that shows that division is not commutative, the general property (using \(a\) and \(b\)) does not exist