{"id":479,"date":"2024-04-19T20:02:06","date_gmt":"2024-04-19T20:02:06","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=479"},"modified":"2025-08-21T22:58:06","modified_gmt":"2025-08-21T22:58:06","slug":"introduction-to-real-numbers-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/introduction-to-real-numbers-learn-it-4\/","title":{"raw":"Introduction to Real Numbers: Learn It 4","rendered":"Introduction to Real Numbers: Learn It 4"},"content":{"raw":"

Properties of Real Numbers Cont.<\/h2>\r\n

Distributive Property<\/h3>\r\nThe distributive property<\/strong> states that the product of a factor times a sum is the sum of the factor times each term in the sum.\r\n
[latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/div>\r\nThis property combines both addition and multiplication (and is the only property to do so).\r\n\r\n
\r\n

distributive property<\/h3>\r\n
\r\n\r\n\r\n\r\n\r\n
Property<\/th>\r\nExample<\/th>\r\nIn Words<\/th>\r\n<\/tr>\r\n
Distributive Property<\/strong>\r\n[latex]a \\times (b + c) = a \\times b + a \\times c[\/latex]<\/td>\r\n[latex]5 \\times (3 + 4) = 5 \\times 3 + 5 \\times 4[\/latex]<\/td>\r\nMultiplication distributes across addition<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/section>
Let us consider an example.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"The Visual example of distribution[\/caption]\r\n\r\nNote that [latex]4[\/latex] is outside the grouping symbols, so we distribute the [latex]4[\/latex] by multiplying it by [latex]12[\/latex], multiplying it by [latex]\u20137[\/latex], and adding the products.<\/section>To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.\r\n
[latex]\\begin{align} 6+\\left(3\\cdot 5\\right)& \\stackrel{?}{=} \\left(6+3\\right)\\cdot \\left(6+5\\right) \\\\ 6+\\left(15\\right)& \\stackrel{?}{=} \\left(9\\right)\\cdot \\left(11\\right) \\\\ 21& \\ne 99 \\end{align}[\/latex]<\/div>\r\nMultiplication does not distribute over subtraction, and division distributes over neither addition nor subtraction.\r\n\r\n
A special case of the distributive property occurs when a sum of terms is subtracted.\r\n
[latex]a-b=a+\\left(-b\\right)[\/latex]<\/div>\r\nFor example, consider the difference [latex]12-\\left(5+3\\right)[\/latex]. We can rewrite the difference of the two terms 12 and [latex]\\left(5+3\\right)[\/latex] by turning the subtraction expression into addition of the opposite. So instead of subtracting [latex]\\left(5+3\\right)[\/latex], we add the opposite.\r\n
[latex]12+\\left(-1\\right)\\cdot \\left(5+3\\right)[\/latex]<\/div>\r\nNow, distribute [latex]-1[\/latex] and simplify the result.\r\n
[latex]\\begin{align}12+\\left(-1\\right)\\cdot\\left(5+3\\right)&=12+[\\left(-1\\right)\\cdot5+\\left(-1\\right)\\cdot3] \\\\&=12+(-5-3) \\\\&=12+\\left(-8\\right) \\\\&=4 \\end{align}[\/latex]<\/div>\r\n

This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.<\/p>\r\n\r\n

[latex]\\begin{align}12-\\left(5+3\\right) &=12+\\left(-5-3\\right) \\\\ &=12+\\left(-8\\right) \\\\ &=4\\end{align}[\/latex]<\/div>\r\n<\/section>
[ohm2_question hide_question_numbers=1]18662[\/ohm2_question]<\/section>
\r\n

Identity Properties<\/h3>\r\nThe identity property of addition<\/strong> states that there is a unique number, called the additive identity ([latex]0[\/latex]) that, when added to a number, results in the original number.\r\n
\u00a0[latex]a+0=a[\/latex]<\/div>\r\nThe identity property of multiplication<\/strong> states that there is a unique number, called the multiplicative identity ([latex]1[\/latex]) that, when multiplied by a number, results in the original number.\r\n
[latex]a\\cdot 1=a[\/latex]<\/div>\r\n<\/section>
\r\n

identity properties<\/h3>\r\n
\r\n\r\n\r\n\r\n\r\n\r\n
Property<\/th>\r\nExample<\/th>\r\nIn Words<\/th>\r\n<\/tr>\r\n
Additive Identity Property<\/strong>\r\n[latex]a + 0 = a[\/latex]<\/td>\r\n[latex]17 + 0 = 17[\/latex]<\/td>\r\nAny number plus [latex]0[\/latex] is the number<\/td>\r\n<\/tr>\r\n
Multiplicative Identity Property<\/strong>\r\n[latex]a \\times 1 = a[\/latex]<\/td>\r\n[latex]21 \\times 1 = 21[\/latex]<\/td>\r\nAny number times one is the number<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/section>
For example, we have [latex]\\left(-6\\right)+0=-6[\/latex] and [latex]23\\cdot 1=23[\/latex].<\/section>There are no exceptions for these properties; they work for every real number, including [latex]0[\/latex] and [latex]1[\/latex].\r\n

Inverse Properties<\/h3>\r\nThe inverse property of addition<\/strong> states that, for every real number a<\/em>, there is a unique number, called the additive inverse (or opposite), denoted\u2212a<\/em>, that, when added to the original number, results in the additive identity, 0.\r\n
[latex]a+\\left(-a\\right)=0[\/latex]<\/div>\r\nFor example, if [latex]a=-8[\/latex], the additive inverse is 8, since [latex]\\left(-8\\right)+8=0[\/latex].\r\n\r\nThe inverse property of multiplication<\/strong> holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number a<\/em>, there is a unique number, called the multiplicative inverse (or reciprocal), denoted [latex]\\frac{1}{a}[\/latex], that, when multiplied by the original number, results in the multiplicative identity, 1.\r\n
[latex]a\\cdot \\dfrac{1}{a}=1[\/latex]<\/div>\r\n

For example, if [latex]a=-\\frac{2}{3}[\/latex], the reciprocal, denoted [latex]\\frac{1}{a}[\/latex], is [latex]-\\frac{3}{2}[\/latex]\u00a0because<\/p>\r\n\r\n

[latex]a\\cdot \\dfrac{1}{a}=\\left(-\\dfrac{2}{3}\\right)\\cdot \\left(-\\dfrac{3}{2}\\right)=1[\/latex]<\/div>\r\n
\r\n

inverse properties<\/h3>\r\n
\r\n
\r\n\r\n\r\n\r\n\r\n\r\n
Property<\/th>\r\nExample<\/th>\r\nIn Words<\/th>\r\n<\/tr>\r\n
Additive Inverse Property<\/strong>\r\n[latex]a + (-a) = 0[\/latex]<\/td>\r\n[latex]14 + (-14) = 0[\/latex]<\/td>\r\nEvery number plus its negative is [latex]0[\/latex]<\/td>\r\n<\/tr>\r\n
Multiplicative Inverse Property<\/strong>\r\n[latex]a \\times \\frac{1}{a} = 1[\/latex], provided [latex](a \\neq 0)[\/latex]<\/td>\r\n[latex]3 \\times \\frac{1}{3} = 1[\/latex]<\/td>\r\nEvery non-zero number times its reciprocal is [latex]1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<\/section>
Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.\r\n
    \r\n \t
  1. [latex]3\\cdot 6+3\\cdot 4[\/latex]<\/li>\r\n \t
  2. [latex]\\left(5+8\\right)+\\left(-8\\right)[\/latex]<\/li>\r\n \t
  3. [latex]6-\\left(15+9\\right)[\/latex]<\/li>\r\n \t
  4. [latex]\\dfrac{4}{7}\\cdot \\left(\\frac{2}{3}\\cdot \\dfrac{7}{4}\\right)[\/latex]<\/li>\r\n \t
  5. [latex]100\\cdot \\left[0.75+\\left(-2.38\\right)\\right][\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"892710\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"892710\"]\r\n\r\n1.\r\n\r\n[latex]\\begin{align}3\\cdot6+3\\cdot4 &=3\\cdot\\left(6+4\\right) && \\text{Distributive property} \\\\ &=3\\cdot10 && \\text{Simplify} \\\\ & =30 && \\text{Simplify}\\end{align}[\/latex]\r\n\r\n2.\r\n\r\n[latex]\\begin{align}\\left(5+8\\right)+\\left(-8\\right) &=5+\\left[8+\\left(-8\\right)\\right] &&\\text{Associative property of addition} \\\\ &=5+0 && \\text{Inverse property of addition} \\\\ &=5 &&\\text{Identity property of addition}\\end{align}[\/latex]\r\n\r\n3.\r\n\r\n[latex]\\begin{align}6-\\left(15+9\\right) & =6+(-15-9) && \\text{Distributive property} \\\\ & =6+\\left(-24\\right) && \\text{Simplify} \\\\ & =-18 && \\text{Simplify}\\end{align}[\/latex]\r\n\r\n4.\r\n\r\n[latex]\\begin{align}\\frac{4}{7}\\cdot\\left(\\frac{2}{3}\\cdot\\frac{7}{4}\\right) & =\\frac{4}{7} \\cdot\\left(\\frac{7}{4}\\cdot\\frac{2}{3}\\right) && \\text{Commutative property of multiplication} \\\\ & =\\left(\\frac{4}{7}\\cdot\\frac{7}{4}\\right)\\cdot\\frac{2}{3} && \\text{Associative property of multiplication} \\\\ & =1\\cdot\\frac{2}{3} && \\text{Inverse property of multiplication} \\\\ & =\\frac{2}{3} && \\text{Identity property of multiplication}\\end{align}[\/latex]\r\n\r\n5.\r\n\r\n[latex]\\begin{align}100\\cdot[0.75+\\left(-2.38\\right)] & =100\\cdot0.75+100\\cdot\\left(-2.38\\right) && \\text{Distributive property} \\\\ & =75+\\left(-238\\right) && \\text{Simplify} \\\\ & =-163 && \\text{Simplify}\\end{align}[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
    [ohm2_question hide_question_numbers=1]12706[\/ohm2_question]<\/section>","rendered":"

    Properties of Real Numbers Cont.<\/h2>\n

    Distributive Property<\/h3>\n

    The distributive property<\/strong> states that the product of a factor times a sum is the sum of the factor times each term in the sum.<\/p>\n

    [latex]a\\cdot \\left(b+c\\right)=a\\cdot b+a\\cdot c[\/latex]<\/div>\n

    This property combines both addition and multiplication (and is the only property to do so).<\/p>\n

    \n

    distributive property<\/h3>\n
    \n\n\n\n\n
    Property<\/th>\nExample<\/th>\nIn Words<\/th>\n<\/tr>\n
    Distributive Property<\/strong>
    \n[latex]a \\times (b + c) = a \\times b + a \\times c[\/latex]<\/td>\n    		[latex]5 \\times (3 + 4) = 5 \\times 3 + 5 \\times 4[\/latex]<\/td>\nMultiplication distributes across addition<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/section>\n
    Let us consider an example.<\/p>\n
    \"The
    Visual example of distribution<\/figcaption><\/figure>\n

    Note that [latex]4[\/latex] is outside the grouping symbols, so we distribute the [latex]4[\/latex] by multiplying it by [latex]12[\/latex], multiplying it by [latex]\u20137[\/latex], and adding the products.<\/section>\n

    To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example.<\/p>\n

    [latex]\\begin{align} 6+\\left(3\\cdot 5\\right)& \\stackrel{?}{=} \\left(6+3\\right)\\cdot \\left(6+5\\right) \\\\ 6+\\left(15\\right)& \\stackrel{?}{=} \\left(9\\right)\\cdot \\left(11\\right) \\\\ 21& \\ne 99 \\end{align}[\/latex]<\/div>\n

    Multiplication does not distribute over subtraction, and division distributes over neither addition nor subtraction.<\/p>\n

    A special case of the distributive property occurs when a sum of terms is subtracted.<\/p>\n
    [latex]a-b=a+\\left(-b\\right)[\/latex]<\/div>\n

    For example, consider the difference [latex]12-\\left(5+3\\right)[\/latex]. We can rewrite the difference of the two terms 12 and [latex]\\left(5+3\\right)[\/latex] by turning the subtraction expression into addition of the opposite. So instead of subtracting [latex]\\left(5+3\\right)[\/latex], we add the opposite.<\/p>\n

    [latex]12+\\left(-1\\right)\\cdot \\left(5+3\\right)[\/latex]<\/div>\n

    Now, distribute [latex]-1[\/latex] and simplify the result.<\/p>\n

    [latex]\\begin{align}12+\\left(-1\\right)\\cdot\\left(5+3\\right)&=12+[\\left(-1\\right)\\cdot5+\\left(-1\\right)\\cdot3] \\\\&=12+(-5-3) \\\\&=12+\\left(-8\\right) \\\\&=4 \\end{align}[\/latex]<\/div>\n

    This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example.<\/p>\n

    [latex]\\begin{align}12-\\left(5+3\\right) &=12+\\left(-5-3\\right) \\\\ &=12+\\left(-8\\right) \\\\ &=4\\end{align}[\/latex]<\/div>\n<\/section>\n