{"id":57,"date":"2023-02-01T00:03:02","date_gmt":"2023-02-01T00:03:02","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/ct-state-quantitative-reasoning\/chapter\/converting-between-bases\/"},"modified":"2023-02-01T00:03:02","modified_gmt":"2023-02-01T00:03:02","slug":"converting-between-bases","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ct-state-quantitative-reasoning\/chapter\/converting-between-bases\/","title":{"raw":"Another Method for Converting Between Bases","rendered":"Another Method for Converting Between Bases"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n \t<li>Use two different methods for converting numbers between bases<\/li>\n<\/ul>\n<\/div>\n<h3>Another Method For Converting From Base 10 to Other Bases<\/h3>\nAs you read the solution to this last example and attempted the \u201cTry It\u201d problems, you may have had to repeatedly stop and think about what was going on. The fact that you are probably struggling to follow the explanation and reproduce the process yourself is mostly due to the fact that the non-decimal systems are so unfamiliar to you. In fact, the only system that you are probably comfortable with is the decimal system.\n\nAs budding mathematicians, you should always be asking questions like \u201cHow could I simplify this process?\u201d In general, that is one of the main things that mathematicians do: they look for ways to take complicated situations and make them easier or more familiar. In this section we will attempt to do that.\n<div class=\"textbox examples\">\n<h3>calculator methods<\/h3>\nThis section presents a method of converting bases that uses a calculator to do the heavy lifting for you. You'll often find, after learning a method to compute a mathematical result by hand, that there is an easier or faster way to do it with a calculator. But it is still beneficial to learn the manual method because the underlying process can contain a logic hidden by the calculator. That logic is often transportable to other situations.\n\n<\/div>\nTo do so, we will start by looking at our own decimal system. What we do may seem obvious and maybe even intuitive but that\u2019s the point. We want to find a process that we readily recognize works and makes sense to us in a familiar system and then use it to extend our results to a different, unfamiliar system.\n\nLet\u2019s start with the decimal number, 4863<sub>10<\/sub>. We will convert this number to base 10. Yeah, I know it\u2019s already in base 10, but if you carefully follow what we\u2019re doing, you\u2019ll see it makes things work out very nicely with other bases later on. We first note that the highest power of 10 that will divide into 4863 at least once is 10<sup>3<\/sup> = 1000. <em>In general, this is the first step in our new process; we find the highest power of a given base that will divide at least once into our given number.<\/em>\n<p style=\"text-align: center;\">We now divide 1000 into 4863:<\/p>\n<p style=\"text-align: center;\">4863 \u00f7 1000 = 4.863<\/p>\nThis says that there are four thousands in 4863 (obviously). However, it also says that there are 0.863 thousands in 4863. This fractional part is our remainder and will be converted to lower powers of our base (10). If we take that decimal and multiply by 10 (since that\u2019s the base we\u2019re in) we get the following:\n<p style=\"text-align: center;\">0.863 \u00d7 10 = 8.63<\/p>\nWhy multiply by 10 at this point? We need to recognize here that 0.863 thousands is the same as 8.63 hundreds. Think about that until it sinks in.\n<p style=\"text-align: center;\">(0.863)(1000) = 863\n(8.63)(100) = 863<\/p>\nThese two statements are equivalent. So, what we are really doing here by multiplying by 10 is rephrasing or converting from one place (thousands) to the next place down (hundreds).\n<p style=\"text-align: center;\">0.863 \u00d7 10 \u21d2 8.63\n(Parts of Thousands) \u00d7 10 \u21d2 Hundreds<\/p>\nWhat we have now is 8 hundreds and a remainder of 0.63 hundreds, which is the same as 6.3 tens. We can do this again with the 0.63 that remains after this first step.\n<p style=\"text-align: center;\">0.63 \u00d7 10 \u21d2 6.3\nHundreds \u00d7 10 \u21d2 Tens<\/p>\nSo we have six tens and 0.3 tens, which is the same as 3 ones, our last place value.\n\nNow here\u2019s the punch line. Let\u2019s put all of the together in one place:\n\n<img class=\"aligncenter wp-image-300 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155212\/Fig5_1_20.png\" alt=\"4863 divided by 1000 = 4.863, 0.863x 10 = 8.63, 0.63x10 = 6.3, 0.3x10 = 3.0\" width=\"237\" height=\"200\">\nNote that in each step, the remainder is carried down to the next step and multiplied by 10, the base. Also, at each step, the whole number part, which is circled, gives the digit that belongs in that particular place. What is amazing is that this works for any base! So, to convert from a base 10 number to some other base, <em>b<\/em>, we have the following steps we can follow:\n<div class=\"textbox\">\n<h3>Converting from Base 10 to Base <em>b<\/em>: Another method<\/h3>\n<ol>\n \t<li>Find the highest power of the base <em>b<\/em> that will divide into the given number at least once and then divide.<\/li>\n \t<li>Keep the whole number part, and multiply the fractional part by the base <em>b<\/em>.<\/li>\n \t<li>Repeat step two, keeping the whole number part (including 0), carrying the fractional part to the next step until only a whole number result is obtained.<\/li>\n \t<li>Collect all your whole number parts to get your number in base <em>b<\/em> notation.<\/li>\n<\/ol>\n<\/div>\nWe will illustrate this procedure with some examples.\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nConvert the base 10 number, 348<sub>10<\/sub>, to base 5.\n[reveal-answer q=\"881622\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"881622\"]\n\nThis is actually a conversion that we have done in a previous example. The powers of five are:\n\n5<sup>0<\/sup> = 1\n5<sup>1<\/sup> = 5\n5<sup>2<\/sup> = 25\n5<sup>3<\/sup> = 125\n5<sup>4<\/sup> = 625\nEtc\u2026\n\nThe highest power of five that will go into 348 at least once is 5<sup>3<\/sup>.\n\nWe divide by 125 and then proceed.\n\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5207\/2020\/04\/09213605\/Screen-Shot-2020-11-09-at-1.35.48-PM.png\"><img class=\"alignnone wp-image-5343 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5207\/2020\/04\/09213605\/Screen-Shot-2020-11-09-at-1.35.48-PM-211x300.png\" alt=\"348\/125=2.784, .784(5)=3.92, .92(5)=4.6, .6(5)=3.0\" width=\"211\" height=\"300\"><\/a>\n\nBy keeping all the whole number parts, from top bottom, gives 2343 as our base 5 number. Thus, 2343<sub>5<\/sub>&nbsp;= 348<sub>10<\/sub>.\n\n&nbsp;\n\n[\/hidden-answer]\n\n<\/div>\nWe can compare our result with what we saw earlier, or simply check with our calculator, and find that these two numbers really are equivalent to each other.\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nConvert the base 10 number, 3007<sub>10<\/sub>, to base 5.\n[reveal-answer q=\"462788\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"462788\"]\n\nThe highest power of 5 that divides at least once into 3007 is 5<sup>4<\/sup> = 625. Thus, we have:\n\n3007 \u00f7 625 = \u2463.8112\n0.8112 \u00d7 5 = \u2463.056\n0.056 \u00d7 5 = \u24ea.28\n0.28 \u00d7 5 = \u24600.4\n0.4 \u00d7 5 = \u24610.0\n\nThis gives us that 3007<sub>10<\/sub> = 44012<sub>5<\/sub>. Notice that in the third line that multiplying by 5 gave us 0 for our whole number part. We don\u2019t discard that! The zero tells us that a zero in that place. That is, there are no 5<sup>2<\/sup>s in this number.\n\n[\/hidden-answer]\n\n<\/div>\nThis last example shows the importance of using a calculator in certain situations and taking care to avoid clearing the calculator\u2019s memory or display until you get to the very end of the process.\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\nConvert the base 10 number, 63201<sub>10<\/sub>, to base 7.\n[reveal-answer q=\"186862\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"186862\"]\n\nThe powers of 7 are:\n\n7<sup>0<\/sup> = 1\n7<sup>1<\/sup> = 7\n7<sup>2<\/sup> = 49\n7<sup>3<\/sup> = 343\n7<sup>4<\/sup> = 2401\n7<sup>5<\/sup> = 16807\netc\u2026\n\nThe highest power of 7 that will divide at least once into 63201 is 7<sup>5<\/sup>. When we do the initial division on a calculator, we get the following:\n\n63201 \u00f7 7<sup>5<\/sup> = 3.760397453\n\nThe decimal part actually fills up the calculators display and we don\u2019t know if it terminates at some point or perhaps even repeats down the road. So if we clear our calculator at this point, we will introduce error that is likely to keep this process from ever ending. To avoid this problem, we leave the result in the calculator and simply subtract 3 from this to get the fractional part all by itself. <strong>Do not round off!<\/strong> Subtraction and then multiplication by seven gives:\n\n63201 \u00f7 7<sup>5<\/sup> = \u2462.760397453\n0.760397453 \u00d7 7 = \u2464.322782174\n0.322782174 \u00d7 7 = \u2461.259475219\n0.259475219 \u00d7 7 = \u2460.816326531\n0.816326531 \u00d7 7 = \u2464.714285714\n0.714285714 \u00d7 7 = \u2464.000000000\n\nYes, believe it or not, that last product is exactly 5, <em>as long as you don\u2019t clear anything out on your calculator<\/em>. This gives us our final result: 63201<sub>10<\/sub> = 352155<sub>7<\/sub>.\n\n[\/hidden-answer]\n\n<\/div>\nIf we round, even to two decimal places in each step, clearing our calculator out at each step along the way, we will get a series of numbers that do not terminate, but begin repeating themselves endlessly. (Try it!) We end up with something that doesn\u2019t make any sense, at least not in this context. So be careful to use your calculator cautiously on these conversion problems.\n\nAlso, remember that if your first division is by 7<sup>5<\/sup>, then you expect to have 6 digits in the final answer, corresponding to the places for 7<sup>5<\/sup>, 7<sup>4<\/sup>, and so on down to 7<sup>0<\/sup>. If you find yourself with more than 6 digits due to rounding errors, you know something went wrong.\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n1. Convert the base 10 number, 9352<sub>10<\/sub>, to base 5.\n[reveal-answer q=\"290694\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"290694\"][latex]9352_{10} = 244402_{5}[\/latex][\/hidden-answer]\n\n2. Convert the base 10 number, 1500, to base 3.\n<p style=\"padding-left: 60px;\">Be careful not to clear your calculator on this one. Also, if you\u2019re not careful in each step, you may not get all of the digits you\u2019re looking for, so move slowly and with caution.<\/p>\n[reveal-answer q=\"148410\"]Show Solution[\/reveal-answer]\n[hidden-answer a=\"148410\"]\n\n[latex]1500_{10} = 2001120_{3}[\/latex]\n\n[\/hidden-answer]\n\n3.\n[ohm_question]8681[\/ohm_question]\n\n<\/div>\nThe following video shows how to use a calculator to convert numbers in base 10 into other bases.\n\nhttps:\/\/youtu.be\/YNPTYelCeIs\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use two different methods for converting numbers between bases<\/li>\n<\/ul>\n<\/div>\n<h3>Another Method For Converting From Base 10 to Other Bases<\/h3>\n<p>As you read the solution to this last example and attempted the \u201cTry It\u201d problems, you may have had to repeatedly stop and think about what was going on. The fact that you are probably struggling to follow the explanation and reproduce the process yourself is mostly due to the fact that the non-decimal systems are so unfamiliar to you. In fact, the only system that you are probably comfortable with is the decimal system.<\/p>\n<p>As budding mathematicians, you should always be asking questions like \u201cHow could I simplify this process?\u201d In general, that is one of the main things that mathematicians do: they look for ways to take complicated situations and make them easier or more familiar. In this section we will attempt to do that.<\/p>\n<div class=\"textbox examples\">\n<h3>calculator methods<\/h3>\n<p>This section presents a method of converting bases that uses a calculator to do the heavy lifting for you. You&#8217;ll often find, after learning a method to compute a mathematical result by hand, that there is an easier or faster way to do it with a calculator. But it is still beneficial to learn the manual method because the underlying process can contain a logic hidden by the calculator. That logic is often transportable to other situations.<\/p>\n<\/div>\n<p>To do so, we will start by looking at our own decimal system. What we do may seem obvious and maybe even intuitive but that\u2019s the point. We want to find a process that we readily recognize works and makes sense to us in a familiar system and then use it to extend our results to a different, unfamiliar system.<\/p>\n<p>Let\u2019s start with the decimal number, 4863<sub>10<\/sub>. We will convert this number to base 10. Yeah, I know it\u2019s already in base 10, but if you carefully follow what we\u2019re doing, you\u2019ll see it makes things work out very nicely with other bases later on. We first note that the highest power of 10 that will divide into 4863 at least once is 10<sup>3<\/sup> = 1000. <em>In general, this is the first step in our new process; we find the highest power of a given base that will divide at least once into our given number.<\/em><\/p>\n<p style=\"text-align: center;\">We now divide 1000 into 4863:<\/p>\n<p style=\"text-align: center;\">4863 \u00f7 1000 = 4.863<\/p>\n<p>This says that there are four thousands in 4863 (obviously). However, it also says that there are 0.863 thousands in 4863. This fractional part is our remainder and will be converted to lower powers of our base (10). If we take that decimal and multiply by 10 (since that\u2019s the base we\u2019re in) we get the following:<\/p>\n<p style=\"text-align: center;\">0.863 \u00d7 10 = 8.63<\/p>\n<p>Why multiply by 10 at this point? We need to recognize here that 0.863 thousands is the same as 8.63 hundreds. Think about that until it sinks in.<\/p>\n<p style=\"text-align: center;\">(0.863)(1000) = 863<br \/>\n(8.63)(100) = 863<\/p>\n<p>These two statements are equivalent. So, what we are really doing here by multiplying by 10 is rephrasing or converting from one place (thousands) to the next place down (hundreds).<\/p>\n<p style=\"text-align: center;\">0.863 \u00d7 10 \u21d2 8.63<br \/>\n(Parts of Thousands) \u00d7 10 \u21d2 Hundreds<\/p>\n<p>What we have now is 8 hundreds and a remainder of 0.63 hundreds, which is the same as 6.3 tens. We can do this again with the 0.63 that remains after this first step.<\/p>\n<p style=\"text-align: center;\">0.63 \u00d7 10 \u21d2 6.3<br \/>\nHundreds \u00d7 10 \u21d2 Tens<\/p>\n<p>So we have six tens and 0.3 tens, which is the same as 3 ones, our last place value.<\/p>\n<p>Now here\u2019s the punch line. Let\u2019s put all of the together in one place:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-300 size-full\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/282\/2016\/01\/20155212\/Fig5_1_20.png\" alt=\"4863 divided by 1000 = 4.863, 0.863x 10 = 8.63, 0.63x10 = 6.3, 0.3x10 = 3.0\" width=\"237\" height=\"200\" \/><br \/>\nNote that in each step, the remainder is carried down to the next step and multiplied by 10, the base. Also, at each step, the whole number part, which is circled, gives the digit that belongs in that particular place. What is amazing is that this works for any base! So, to convert from a base 10 number to some other base, <em>b<\/em>, we have the following steps we can follow:<\/p>\n<div class=\"textbox\">\n<h3>Converting from Base 10 to Base <em>b<\/em>: Another method<\/h3>\n<ol>\n<li>Find the highest power of the base <em>b<\/em> that will divide into the given number at least once and then divide.<\/li>\n<li>Keep the whole number part, and multiply the fractional part by the base <em>b<\/em>.<\/li>\n<li>Repeat step two, keeping the whole number part (including 0), carrying the fractional part to the next step until only a whole number result is obtained.<\/li>\n<li>Collect all your whole number parts to get your number in base <em>b<\/em> notation.<\/li>\n<\/ol>\n<\/div>\n<p>We will illustrate this procedure with some examples.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Convert the base 10 number, 348<sub>10<\/sub>, to base 5.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q881622\">Show Solution<\/span><\/p>\n<div id=\"q881622\" class=\"hidden-answer\" style=\"display: none\">\n<p>This is actually a conversion that we have done in a previous example. The powers of five are:<\/p>\n<p>5<sup>0<\/sup> = 1<br \/>\n5<sup>1<\/sup> = 5<br \/>\n5<sup>2<\/sup> = 25<br \/>\n5<sup>3<\/sup> = 125<br \/>\n5<sup>4<\/sup> = 625<br \/>\nEtc\u2026<\/p>\n<p>The highest power of five that will go into 348 at least once is 5<sup>3<\/sup>.<\/p>\n<p>We divide by 125 and then proceed.<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5207\/2020\/04\/09213605\/Screen-Shot-2020-11-09-at-1.35.48-PM.png\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-5343 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/5207\/2020\/04\/09213605\/Screen-Shot-2020-11-09-at-1.35.48-PM-211x300.png\" alt=\"348\/125=2.784, .784(5)=3.92, .92(5)=4.6, .6(5)=3.0\" width=\"211\" height=\"300\" \/><\/a><\/p>\n<p>By keeping all the whole number parts, from top bottom, gives 2343 as our base 5 number. Thus, 2343<sub>5<\/sub>&nbsp;= 348<sub>10<\/sub>.<\/p>\n<p>&nbsp;<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>We can compare our result with what we saw earlier, or simply check with our calculator, and find that these two numbers really are equivalent to each other.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Convert the base 10 number, 3007<sub>10<\/sub>, to base 5.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q462788\">Show Solution<\/span><\/p>\n<div id=\"q462788\" class=\"hidden-answer\" style=\"display: none\">\n<p>The highest power of 5 that divides at least once into 3007 is 5<sup>4<\/sup> = 625. Thus, we have:<\/p>\n<p>3007 \u00f7 625 = \u2463.8112<br \/>\n0.8112 \u00d7 5 = \u2463.056<br \/>\n0.056 \u00d7 5 = \u24ea.28<br \/>\n0.28 \u00d7 5 = \u24600.4<br \/>\n0.4 \u00d7 5 = \u24610.0<\/p>\n<p>This gives us that 3007<sub>10<\/sub> = 44012<sub>5<\/sub>. Notice that in the third line that multiplying by 5 gave us 0 for our whole number part. We don\u2019t discard that! The zero tells us that a zero in that place. That is, there are no 5<sup>2<\/sup>s in this number.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>This last example shows the importance of using a calculator in certain situations and taking care to avoid clearing the calculator\u2019s memory or display until you get to the very end of the process.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Convert the base 10 number, 63201<sub>10<\/sub>, to base 7.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q186862\">Show Solution<\/span><\/p>\n<div id=\"q186862\" class=\"hidden-answer\" style=\"display: none\">\n<p>The powers of 7 are:<\/p>\n<p>7<sup>0<\/sup> = 1<br \/>\n7<sup>1<\/sup> = 7<br \/>\n7<sup>2<\/sup> = 49<br \/>\n7<sup>3<\/sup> = 343<br \/>\n7<sup>4<\/sup> = 2401<br \/>\n7<sup>5<\/sup> = 16807<br \/>\netc\u2026<\/p>\n<p>The highest power of 7 that will divide at least once into 63201 is 7<sup>5<\/sup>. When we do the initial division on a calculator, we get the following:<\/p>\n<p>63201 \u00f7 7<sup>5<\/sup> = 3.760397453<\/p>\n<p>The decimal part actually fills up the calculators display and we don\u2019t know if it terminates at some point or perhaps even repeats down the road. So if we clear our calculator at this point, we will introduce error that is likely to keep this process from ever ending. To avoid this problem, we leave the result in the calculator and simply subtract 3 from this to get the fractional part all by itself. <strong>Do not round off!<\/strong> Subtraction and then multiplication by seven gives:<\/p>\n<p>63201 \u00f7 7<sup>5<\/sup> = \u2462.760397453<br \/>\n0.760397453 \u00d7 7 = \u2464.322782174<br \/>\n0.322782174 \u00d7 7 = \u2461.259475219<br \/>\n0.259475219 \u00d7 7 = \u2460.816326531<br \/>\n0.816326531 \u00d7 7 = \u2464.714285714<br \/>\n0.714285714 \u00d7 7 = \u2464.000000000<\/p>\n<p>Yes, believe it or not, that last product is exactly 5, <em>as long as you don\u2019t clear anything out on your calculator<\/em>. This gives us our final result: 63201<sub>10<\/sub> = 352155<sub>7<\/sub>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>If we round, even to two decimal places in each step, clearing our calculator out at each step along the way, we will get a series of numbers that do not terminate, but begin repeating themselves endlessly. (Try it!) We end up with something that doesn\u2019t make any sense, at least not in this context. So be careful to use your calculator cautiously on these conversion problems.<\/p>\n<p>Also, remember that if your first division is by 7<sup>5<\/sup>, then you expect to have 6 digits in the final answer, corresponding to the places for 7<sup>5<\/sup>, 7<sup>4<\/sup>, and so on down to 7<sup>0<\/sup>. If you find yourself with more than 6 digits due to rounding errors, you know something went wrong.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>1. Convert the base 10 number, 9352<sub>10<\/sub>, to base 5.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q290694\">Show Solution<\/span><\/p>\n<div id=\"q290694\" class=\"hidden-answer\" style=\"display: none\">[latex]9352_{10} = 244402_{5}[\/latex]<\/div>\n<\/div>\n<p>2. Convert the base 10 number, 1500, to base 3.<\/p>\n<p style=\"padding-left: 60px;\">Be careful not to clear your calculator on this one. Also, if you\u2019re not careful in each step, you may not get all of the digits you\u2019re looking for, so move slowly and with caution.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q148410\">Show Solution<\/span><\/p>\n<div id=\"q148410\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]1500_{10} = 2001120_{3}[\/latex]<\/p>\n<\/div>\n<\/div>\n<p>3.<br \/>\n<iframe loading=\"lazy\" id=\"ohm8681\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=8681&theme=oea&iframe_resize_id=ohm8681&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>The following video shows how to use a calculator to convert numbers in base 10 into other bases.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Convert Numbers in Base Ten to Different Bases:  Calculator Method\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/YNPTYelCeIs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-57\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Math in Society. <strong>Authored by<\/strong>: Lippman, David. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\">http:\/\/www.opentextbookstore.com\/mathinsociety\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Question ID 8681. <strong>Authored by<\/strong>: Lippman, David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li> Convert Numbers in Base Ten to Different Bases:  Calculator Method. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/YNPTYelCeIs\">https:\/\/youtu.be\/YNPTYelCeIs<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":538461,"menu_order":24,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Math in Society\",\"author\":\"Lippman, David\",\"organization\":\"\",\"url\":\"http:\/\/www.opentextbookstore.com\/mathinsociety\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Question ID 8681\",\"author\":\"Lippman, David\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"},{\"type\":\"cc\",\"description\":\" Convert Numbers in Base Ten to Different Bases:  Calculator Method\",\"author\":\"James Sousa (Mathispower4u.com) \",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/YNPTYelCeIs\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and 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