reciprocal<\/h3>\r\n
The reciprocal of a fraction [latex]\\frac{a}{b}[\/latex] is [latex]\\frac{b}{a}[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are non-zero. Multiplying a fraction by its reciprocal always results in [latex]1[\/latex].<\/p>\r\n<\/div>\r\n<\/section>\r\n
If you multiply two numbers together and get [latex]1[\/latex] as a result, then the two numbers are reciprocals. Here are some examples of reciprocals:<\/p>\r\n
| Original number<\/th>\r\n | Reciprocal<\/th>\r\n | Product<\/th>\r\n<\/tr>\r\n<\/thead>\r\n | |||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| [latex] \\frac{3}{4}[\/latex]<\/td>\r\n | [latex] \\frac{4}{3}[\/latex]<\/td>\r\n | [latex] \\frac{3}{4} \\times \\frac{4}{3}=\\frac{3 \\times 4}{4 \\times 3}=\\frac{12}{12}=1[\/latex]<\/td>\r\n<\/tr>\r\n | |||||||||||||||
| [latex] \\frac{1}{2}[\/latex]<\/td>\r\n | [latex] \\frac{2}{1}[\/latex]<\/td>\r\n | [latex]\\frac{1}{2} \\times \\frac{2}{1}=\\frac{1 \\times 2}{2 \\times 1}=\\frac{2}{2}=1[\/latex]<\/td>\r\n<\/tr>\r\n | |||||||||||||||
| [latex] 3=\\frac{3}{1}[\/latex]<\/td>\r\n | [latex] \\frac{1}{3}[\/latex]<\/td>\r\n | [latex] \\frac{3}{1} \\times \\frac{1}{3}=\\frac{3 \\times 1}{1 \\times 3}=\\frac{3}{3}=1[\/latex]<\/td>\r\n<\/tr>\r\n | |||||||||||||||
| [latex]2\\frac{1}{3}=\\frac{7}{3}[\/latex]<\/td>\r\n | [latex] \\frac{3}{7}[\/latex]<\/td>\r\n | [latex]\\frac{7}{3} \\times \\frac{3}{7}=\\frac{7 \\times 3}{3 \\times 7}=\\frac{21}{21}=1[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n Sometimes we call\u00a0the reciprocal\u00a0the \u201cflip\u201d of the other number: flip [latex] \\frac{2}{5}[\/latex] to get the reciprocal [latex]\\frac{5}{2}[\/latex].<\/p>\r\n When dividing two rational numbers, find the reciprocal of the divisor (the number that is being divided into the other number). Next, replace the divisor by its reciprocal and change the division into multiplication. Then, perform the multiplication.<\/p>\r\n Dividing rational numbers involves a process where you multiply by the reciprocal of the number you\u2019re dividing by.<\/p>\r\n <\/p>\r\n Symbolically, we write this as: If [latex]b[\/latex], [latex]c[\/latex] and [latex]d[\/latex] are non-zero integers, then<\/p>\r\n Any easy way to remember how to divide fractions is the phrase \u201ckeep, change, flip.\u201d This means to KEEP<\/strong> the first number, CHANGE<\/strong> the division sign to multiplication, and then FLIP<\/strong> (use the reciprocal) of the second number.<\/p>\r\n<\/section>\r\n KEEP<\/strong> [latex] \\frac{2}{3}[\/latex] CHANGE<\/strong>\u00a0 [latex] \\div [\/latex] to \u00a0[latex] \\times [\/latex] FLIP\u00a0<\/strong> [latex]\\frac{1}{6}[\/latex]<\/p>\r\n [latex] \\frac{2}{3} \\times \\frac{6}{1}[\/latex]<\/p>\r\n Multiply numerators and multiply denominators.<\/p>\r\n [latex]\\frac{2 \\times 6}{3 \\times 1}=\\frac{12}{3}[\/latex]<\/p>\r\n Simplify.<\/p>\r\n [latex] \\frac{12}{3}=4[\/latex]<\/p>\r\n [ohm2_question hide_question_numbers=1]12696[\/ohm2_question]<\/p>\r\n<\/section>\r\n You know what it means to divide by [latex]2[\/latex] or divide by [latex]10[\/latex], but what does it mean to divide a quantity by [latex]0[\/latex]? Is this even possible? Can you divide 0 by a number? Consider the fraction<\/p>\r\n [latex]\\frac{0}{8}[\/latex]<\/p>\r\n We can read it as, \u201czero divided by eight.\u201d Since multiplication is the inverse of division, we could rewrite this as a multiplication problem.<\/p>\r\n [latex]\\text{?}\\cdot{8}=0[\/latex].<\/p>\r\n We can infer that the unknown must be [latex]0[\/latex] since that is the only number that will give a result of [latex]0[\/latex] when it is multiplied by [latex]8[\/latex].<\/p>\r\n Now let\u2019s consider the reciprocal of [latex]\\frac{0}{8}[\/latex] which would be [latex]\\frac{8}{0}[\/latex]. If we\u00a0rewrite this as a multiplication problem, we will have<\/p>\r\n [latex]\\text{?}\\cdot{0}=8[\/latex].<\/p>\r\n This doesn't make any sense. There are no numbers that you can multiply by zero to get a result of [latex]8[\/latex]. The reciprocal of [latex]\\frac{8}{0}[\/latex] is undefined, and in fact, all division by zero is undefined.<\/p>\r\n Before discussing division of rational numbers, we should look at the reciprocal <\/strong>of a number. The reciprocal of a number is [latex]1[\/latex] divided by the number. For a fraction, the reciprocal is the fraction formed by switching the numerator and denominator.\u00a0 An important feature for a number and its reciprocal is that their product is [latex]1[\/latex].<\/p>\n The reciprocal of a fraction [latex]\\frac{a}{b}[\/latex] is [latex]\\frac{b}{a}[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are non-zero. Multiplying a fraction by its reciprocal always results in [latex]1[\/latex].<\/p>\n<\/div>\n<\/section>\n If you multiply two numbers together and get [latex]1[\/latex] as a result, then the two numbers are reciprocals. Here are some examples of reciprocals:<\/p>\n Sometimes we call\u00a0the reciprocal\u00a0the \u201cflip\u201d of the other number: flip [latex]\\frac{2}{5}[\/latex] to get the reciprocal [latex]\\frac{5}{2}[\/latex].<\/p>\n When dividing two rational numbers, find the reciprocal of the divisor (the number that is being divided into the other number). Next, replace the divisor by its reciprocal and change the division into multiplication. Then, perform the multiplication.<\/p>\n Dividing rational numbers involves a process where you multiply by the reciprocal of the number you\u2019re dividing by.<\/p>\n <\/p>\n Symbolically, we write this as: If [latex]b[\/latex], [latex]c[\/latex] and [latex]d[\/latex] are non-zero integers, then<\/p>\n Any easy way to remember how to divide fractions is the phrase \u201ckeep, change, flip.\u201d This means to KEEP<\/strong> the first number, CHANGE<\/strong> the division sign to multiplication, and then FLIP<\/strong> (use the reciprocal) of the second number.<\/p>\n<\/section>\n |
![\"Caution\"](\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\")
Caution! Division by zero is undefined and so is the reciprocal of any fraction that has a zero in the numerator. For any real number [latex]a[\/latex], [latex]\\frac{a}{0}[\/latex] is undefined. Additionally, the reciprocal of [latex]\\frac{0}{a}[\/latex] will always be undefined.<\/p>\r\n<\/div>","rendered":"