{"id":473,"date":"2024-04-19T19:34:53","date_gmt":"2024-04-19T19:34:53","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=473"},"modified":"2024-11-20T00:51:58","modified_gmt":"2024-11-20T00:51:58","slug":"introduction-to-real-numbers-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/introduction-to-real-numbers-learn-it-3\/","title":{"raw":"Introduction to Real Numbers: Learn It 3","rendered":"Introduction to Real Numbers: Learn It 3"},"content":{"raw":"

Properties of Real Numbers<\/h2>\r\nFor some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.\r\n

Commutative Properties<\/h3>\r\nThe commutative property of addition<\/strong> states that numbers may be added in any order without affecting the sum.\r\n
[latex]a+b=b+a[\/latex]<\/div>\r\n
For example: [latex]\\left(-2\\right)+7=5\\text{ and }7+\\left(-2\\right)=5[\/latex]<\/div>\r\n \r\n\r\nSimilarly, the commutative property of multiplication<\/strong> states that numbers may be multiplied in any order without affecting the product.\r\n
[latex]a\\cdot b=b\\cdot a[\/latex]<\/div>\r\n
For example: [latex]\\left(-11\\right)\\cdot\\left(-4\\right)=44\\text{ and }\\left(-4\\right)\\cdot\\left(-11\\right)=44[\/latex]<\/div>\r\n
\r\n

commutative 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
Commutative Property of Addition<\/strong>\r\n[latex]a + b = b + a[\/latex]<\/td>\r\n[latex]3 + 7 = 7 + 3[\/latex]<\/td>\r\nNumbers can be added in any order<\/td>\r\n<\/tr>\r\n
Commutative Property of Multiplication<\/strong>\r\n[latex]a \\times b = b \\times a[\/latex]<\/td>\r\n[latex]10 \\times 4 = 4 \\times 10[\/latex]<\/td>\r\nNumbers can be multiplied in any order<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/section>
It is important to note that neither subtraction nor division is commutative. For example, [latex]17 - 5[\/latex] is not the same as [latex]5 - 17[\/latex]. Similarly, [latex]20\\div 5\\ne 5\\div 20[\/latex].<\/section>
[ohm2_question hide_question_numbers=1]18657[\/ohm2_question]<\/section>
[ohm2_question hide_question_numbers=1]18658[\/ohm2_question]<\/section>
\r\n

Associative Properties<\/h3>\r\nThe associative property of multiplication<\/strong> tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.\r\n
[latex]a\\left(bc\\right)=\\left(ab\\right)c[\/latex]<\/div>\r\nConsider this example: [latex]\\left(3\\cdot4\\right)\\cdot5=60\\text{ and }3\\cdot\\left(4\\cdot5\\right)=60[\/latex]\r\n\r\n \r\n\r\nThe associative property of addition<\/strong> tells us that numbers may be grouped differently without affecting the sum.\r\n
[latex]a+\\left(b+c\\right)=\\left(a+b\\right)+c[\/latex]<\/div>\r\nThis property can be especially helpful when dealing with negative integers.\r\n\r\nConsider this example: [latex][15+\\left(-9\\right)]+23=29\\text{ and }15+[\\left(-9\\right)+23]=29[\/latex]\r\n\r\n<\/section>
\r\n

associative properties<\/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
Associative Property of Addition<\/strong>\r\n[latex]a + (b + c) = (a + b) + c[\/latex]<\/td>\r\n[latex]4 + (3 + 8) = (4 + 3) + 8[\/latex]<\/td>\r\nDoesn't matter which pair of numbers is added first<\/td>\r\n<\/tr>\r\n
Associative Property of Multiplication<\/strong>\r\n[latex]a \\times (b \\times c) = (a \\times b) \\times c[\/latex]<\/td>\r\n[latex]2 \\times (5 \\times 7) = (2 \\times 5) \\times 7[\/latex]<\/td>\r\nDoesn't matter which pair of numbers is multiplied first<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section>
Are subtraction and division associative?To answer this, review these examples.\r\n
[latex]\\begin{align}8-\\left(3-15\\right) & \\stackrel{?}{=}\\left(8-3\\right)-15 \\\\ 8-\\left(-12\\right) & \\stackrel{?}=5-15 \\\\ 20 & \\neq 20-10 \\\\ \\text{ }\\end{align}[\/latex]<\/div>\r\n
[latex]\\begin{align}64\\div\\left(8\\div4\\right)&\\stackrel{?}{=}\\left(64\\div8\\right)\\div4 \\\\ 64\\div2 & \\stackrel{?}{=}8\\div4 \\\\ 32 & \\neq 2 \\\\ \\text{ }\\end{align}[\/latex]<\/div>\r\n
As we can see, neither subtraction nor division is associative.<\/div>\r\n<\/section>
[ohm2_question hide_question_numbers=1]18659[\/ohm2_question]<\/section>
[ohm2_question hide_question_numbers=1]18660[\/ohm2_question]<\/section>","rendered":"

Properties of Real Numbers<\/h2>\n

For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.<\/p>\n

Commutative Properties<\/h3>\n

The commutative property of addition<\/strong> states that numbers may be added in any order without affecting the sum.<\/p>\n

[latex]a+b=b+a[\/latex]<\/div>\n
For example: [latex]\\left(-2\\right)+7=5\\text{ and }7+\\left(-2\\right)=5[\/latex]<\/div>\n

 <\/p>\n

Similarly, the commutative property of multiplication<\/strong> states that numbers may be multiplied in any order without affecting the product.<\/p>\n

[latex]a\\cdot b=b\\cdot a[\/latex]<\/div>\n
For example: [latex]\\left(-11\\right)\\cdot\\left(-4\\right)=44\\text{ and }\\left(-4\\right)\\cdot\\left(-11\\right)=44[\/latex]<\/div>\n
\n

commutative properties<\/h3>\n
\n\n\n\n\n\n
Property<\/th>\nExample<\/th>\nIn Words<\/th>\n<\/tr>\n
Commutative Property of Addition<\/strong>
\n[latex]a + b = b + a[\/latex]<\/td>\n		[latex]3 + 7 = 7 + 3[\/latex]<\/td>\nNumbers can be added in any order<\/td>\n<\/tr>\n
Commutative Property of Multiplication<\/strong>
\n[latex]a \\times b = b \\times a[\/latex]<\/td>\n		[latex]10 \\times 4 = 4 \\times 10[\/latex]<\/td>\nNumbers can be multiplied in any order<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/section>\n
It is important to note that neither subtraction nor division is commutative. For example, [latex]17 - 5[\/latex] is not the same as [latex]5 - 17[\/latex]. Similarly, [latex]20\\div 5\\ne 5\\div 20[\/latex].<\/section>\n