{"id":51,"date":"2023-01-25T16:33:56","date_gmt":"2023-01-25T16:33:56","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/positional-systems-and-bases-learn-it-page-2\/"},"modified":"2024-10-18T20:52:29","modified_gmt":"2024-10-18T20:52:29","slug":"positional-systems-and-bases-learn-it-2","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/positional-systems-and-bases-learn-it-2\/","title":{"raw":"Positional Systems and Bases: Learn It 2","rendered":"Positional Systems and Bases: Learn It 2"},"content":{"raw":"

Converting from Other Bases to Base [latex]10[\/latex]<\/h2>\r\n

In the previous sections, we have been referring to positional base systems. In this section of the module, we will explore exactly what a base system is and what it means if a system is \u201cpositional.\u201d<\/p>\r\n

A base system<\/strong> is a structure within which we count. The easiest way to describe a base system is to think about our own base-[latex]10[\/latex] system. The base-[latex]10[\/latex] system, which we call the \u201cdecimal\u201d system, requires a total of ten different symbols\/digits to write any number. They are, of course, [latex]0[\/latex], [latex]1[\/latex], [latex]2[\/latex], . . . , [latex]9[\/latex].<\/p>\r\n

The decimal system is also an example of a positional <\/em>base system, which simply means that the position of a digit gives its place value. Not all civilizations had a positional system even though they did have a base with which they worked.<\/p>\r\n

\r\n
\r\n

base system<\/h3>\r\n

Base systems<\/strong>, also known as numeral systems, are ways of representing numbers that rely on a specific 'base' or radix to structure the notation.<\/p>\r\n<\/div>\r\n<\/section>\r\n

\r\n

The most common base system is decimal or base-[latex]10[\/latex], which uses ten digits ([latex]0-9[\/latex]), but other systems like binary (base-[latex]2[\/latex]), octal (base-[latex]8[\/latex]), and hexadecimal (base-[latex]16[\/latex]) are often used in computing and digital technologies.<\/p>\r\n<\/section>\r\n

Although it is the base our number system uses, base-[latex]10[\/latex] is not the only option we have. Practically any positive integer greater than or equal to [latex]2[\/latex] can be used as a base for a number system. Such systems can work just like the decimal system except the number of symbols will be different and each position will depend on the base itself.<\/p>\r\n

For example, let\u2019s suppose we adopt a base-five system. The only modern digits we would need for this system are [latex]0[\/latex], [latex]1[\/latex], [latex]2[\/latex], [latex]3[\/latex] and [latex]4[\/latex]. What are the place values in such a system? To answer that, we start with the ones place, as most base systems do. However, if we were to count in this system, we could only get to four [latex](4)[\/latex] before we had to jump up to the next place. Our base is [latex]5[\/latex], after all! What is that next place that we would jump to? It would not be tens, since we are no longer in base-[latex]10[\/latex]. We\u2019re in a different numerical world. As the base-[latex]10[\/latex] system progresses from [latex]10^0[\/latex] to [latex]10^1[\/latex], so the base-five system moves from [latex]5^0[\/latex] to [latex]5^1 = 5[\/latex]. Thus, we move from the ones to the fives.<\/p>\r\n

After the fives, we would move to the [latex]5^2[\/latex] place, or the twenty-fives. Note that in base-[latex]10[\/latex], we would have gone from the tens to the hundreds, which is, of course, [latex]10^2[\/latex].<\/p>\r\n

\r\n

Let\u2019s try an example and build a table.<\/p>\r\n

Consider the number [latex]30412[\/latex] in base five. We will write this as [latex]30412_5[\/latex], where the subscript [latex]5[\/latex] is not part of the number but indicates the base we\u2019re using.<\/p>\r\n

First off, note that this is NOT the number \u201cthirty thousand, four hundred twelve.\u201d We must be careful not to impose the base-[latex]10[\/latex] system on this number.<\/p>\r\n

Here\u2019s what our table might look like. We will use it to convert this number to our more familiar base-[latex]10[\/latex] system.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
 <\/td>\r\nBase-[latex]5[\/latex]<\/td>\r\nThis column coverts to Base-[latex]10[\/latex]<\/td>\r\nIn Base-[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n
 <\/td>\r\n[latex]3 \u00d7 5^4[\/latex]<\/td>\r\n[latex]= 3 \u00d7 625[\/latex]<\/td>\r\n[latex]= 1875[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]+[\/latex]<\/td>\r\n[latex]0 \u00d7 5^3[\/latex]<\/td>\r\n[latex]= 0 \u00d7 125[\/latex]<\/td>\r\n[latex]= 0[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]+[\/latex]<\/td>\r\n[latex]4 \u00d7 5^2[\/latex]<\/td>\r\n[latex]= 4 \u00d7 25[\/latex]<\/td>\r\n[latex]= 100[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]+[\/latex]<\/td>\r\n[latex]1 \u00d7 5^1[\/latex]<\/td>\r\n[latex]= 1 \u00d7 5[\/latex]<\/td>\r\n[latex]= 5[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]+[\/latex]<\/td>\r\n[latex]2 \u00d7 5^0[\/latex]<\/td>\r\n[latex]= 2 \u00d7 1[\/latex]<\/td>\r\n[latex]= 2[\/latex]<\/td>\r\n<\/tr>\r\n
 <\/td>\r\n <\/td>\r\nTotal:\u00a0<\/td>\r\n[latex]1982[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

 <\/p>\r\n

As you can see, the number [latex]30412_5[\/latex] is equivalent to [latex]1982[\/latex] in base-[latex]10[\/latex]. We will say [latex]30412_5 = 1982_{10}[\/latex]. All of this may seem strange to you, but that\u2019s only because you are so used to the only system that you\u2019ve probably ever seen.<\/p>\r\n<\/section>\r\n

\r\n

How to: Convert from Base-[latex]b[\/latex] to Base-[latex]10[\/latex]<\/strong><\/p>\r\n

    \r\n\t
  1. Identify the place value for each digit in the base-[latex]b[\/latex] number, starting from the rightmost place as the \"units\" place (equivalent to base-[latex]b^0[\/latex]), and then increasing the power by one with each position to the left.<\/li>\r\n\t
  2. Multiply each digit of the base-[latex]b[\/latex] number by the corresponding base-[latex]b[\/latex] place value.<\/li>\r\n\t
  3. Sum up the values obtained in the second step, which results in the equivalent base-[latex]10[\/latex] number.<\/li>\r\n\t
  4. Proceed in this way, from right to left, until every digit of the base-[latex]b[\/latex] number has been processed.<\/li>\r\n\t
  5. The total sum after all operations is your converted base-[latex]10[\/latex] number.<\/li>\r\n<\/ol>\r\n<\/section>\r\n
    Convert [latex]6234_{7}[\/latex]\u00a0to a base-[latex]10[\/latex] number.
    \r\n[reveal-answer q=\"482364\"]Show Solution[\/reveal-answer]
    \r\n[hidden-answer a=\"482364\"]We first note that we are given a base-[latex]7[\/latex] number that we are to convert. Thus, our places will start at the ones ([latex]7^{0}[\/latex]), and then move up to the [latex]7[\/latex]s, [latex]49[\/latex]s ([latex]7^{2}[\/latex]), etc. Here\u2019s the breakdown:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
     <\/td>\r\nBase-[latex]7[\/latex]<\/td>\r\nConvert<\/td>\r\nBase-[latex]10[\/latex]<\/td>\r\n<\/tr>\r\n
     <\/td>\r\n[latex]= 6 \u00d7 7^{3}[\/latex]<\/td>\r\n[latex]= 6\u00a0\u00d7 343[\/latex]<\/td>\r\n[latex]= 2058[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]+[\/latex]<\/td>\r\n[latex]= 2 \u00d7 7^{2}[\/latex]<\/td>\r\n[latex]= 2\u00a0\u00d7 49[\/latex]<\/td>\r\n[latex]= 98[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]+[\/latex]<\/td>\r\n[latex]= 3 \u00d7 7^{1}[\/latex]<\/td>\r\n[latex]= 3\u00a0\u00d7 7[\/latex]<\/td>\r\n[latex]= 21[\/latex]<\/td>\r\n<\/tr>\r\n
    [latex]+[\/latex]<\/td>\r\n[latex]= 4 \u00d7 7^{0}[\/latex]<\/td>\r\n[latex]= 4\u00a0\u00d7 1[\/latex]<\/td>\r\n[latex]= 4[\/latex]<\/td>\r\n<\/tr>\r\n
     <\/td>\r\n <\/td>\r\nTotal:\u00a0<\/td>\r\n[latex]2181[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

     <\/p>\r\n

    Thus [latex]6234_{7}=2181_{10}[\/latex]<\/p>\r\n

    [\/hidden-answer]<\/p>\r\n<\/section>\r\n

    [ohm2_question hide_question_numbers=1]2331[\/ohm2_question]<\/section>","rendered":"

    Converting from Other Bases to Base [latex]10[\/latex]<\/h2>\n

    In the previous sections, we have been referring to positional base systems. In this section of the module, we will explore exactly what a base system is and what it means if a system is \u201cpositional.\u201d<\/p>\n

    A base system<\/strong> is a structure within which we count. The easiest way to describe a base system is to think about our own base-[latex]10[\/latex] system. The base-[latex]10[\/latex] system, which we call the \u201cdecimal\u201d system, requires a total of ten different symbols\/digits to write any number. They are, of course, [latex]0[\/latex], [latex]1[\/latex], [latex]2[\/latex], . . . , [latex]9[\/latex].<\/p>\n

    The decimal system is also an example of a positional <\/em>base system, which simply means that the position of a digit gives its place value. Not all civilizations had a positional system even though they did have a base with which they worked.<\/p>\n

    \n
    \n

    base system<\/h3>\n

    Base systems<\/strong>, also known as numeral systems, are ways of representing numbers that rely on a specific ‘base’ or radix to structure the notation.<\/p>\n<\/div>\n<\/section>\n

    \n

    The most common base system is decimal or base-[latex]10[\/latex], which uses ten digits ([latex]0-9[\/latex]), but other systems like binary (base-[latex]2[\/latex]), octal (base-[latex]8[\/latex]), and hexadecimal (base-[latex]16[\/latex]) are often used in computing and digital technologies.<\/p>\n<\/section>\n

    Although it is the base our number system uses, base-[latex]10[\/latex] is not the only option we have. Practically any positive integer greater than or equal to [latex]2[\/latex] can be used as a base for a number system. Such systems can work just like the decimal system except the number of symbols will be different and each position will depend on the base itself.<\/p>\n

    For example, let\u2019s suppose we adopt a base-five system. The only modern digits we would need for this system are [latex]0[\/latex], [latex]1[\/latex], [latex]2[\/latex], [latex]3[\/latex] and [latex]4[\/latex]. What are the place values in such a system? To answer that, we start with the ones place, as most base systems do. However, if we were to count in this system, we could only get to four [latex](4)[\/latex] before we had to jump up to the next place. Our base is [latex]5[\/latex], after all! What is that next place that we would jump to? It would not be tens, since we are no longer in base-[latex]10[\/latex]. We\u2019re in a different numerical world. As the base-[latex]10[\/latex] system progresses from [latex]10^0[\/latex] to [latex]10^1[\/latex], so the base-five system moves from [latex]5^0[\/latex] to [latex]5^1 = 5[\/latex]. Thus, we move from the ones to the fives.<\/p>\n

    After the fives, we would move to the [latex]5^2[\/latex] place, or the twenty-fives. Note that in base-[latex]10[\/latex], we would have gone from the tens to the hundreds, which is, of course, [latex]10^2[\/latex].<\/p>\n

    \n

    Let\u2019s try an example and build a table.<\/p>\n

    Consider the number [latex]30412[\/latex] in base five. We will write this as [latex]30412_5[\/latex], where the subscript [latex]5[\/latex] is not part of the number but indicates the base we\u2019re using.<\/p>\n

    First off, note that this is NOT the number \u201cthirty thousand, four hundred twelve.\u201d We must be careful not to impose the base-[latex]10[\/latex] system on this number.<\/p>\n

    Here\u2019s what our table might look like. We will use it to convert this number to our more familiar base-[latex]10[\/latex] system.<\/p>\n\n\n\n\n\n\n\n\n\n
     <\/td>\nBase-[latex]5[\/latex]<\/td>\nThis column coverts to Base-[latex]10[\/latex]<\/td>\nIn Base-[latex]10[\/latex]<\/td>\n<\/tr>\n
     <\/td>\n[latex]3 \u00d7 5^4[\/latex]<\/td>\n[latex]= 3 \u00d7 625[\/latex]<\/td>\n[latex]= 1875[\/latex]<\/td>\n<\/tr>\n
    [latex]+[\/latex]<\/td>\n[latex]0 \u00d7 5^3[\/latex]<\/td>\n[latex]= 0 \u00d7 125[\/latex]<\/td>\n[latex]= 0[\/latex]<\/td>\n<\/tr>\n
    [latex]+[\/latex]<\/td>\n[latex]4 \u00d7 5^2[\/latex]<\/td>\n[latex]= 4 \u00d7 25[\/latex]<\/td>\n[latex]= 100[\/latex]<\/td>\n<\/tr>\n
    [latex]+[\/latex]<\/td>\n[latex]1 \u00d7 5^1[\/latex]<\/td>\n[latex]= 1 \u00d7 5[\/latex]<\/td>\n[latex]= 5[\/latex]<\/td>\n<\/tr>\n
    [latex]+[\/latex]<\/td>\n[latex]2 \u00d7 5^0[\/latex]<\/td>\n[latex]= 2 \u00d7 1[\/latex]<\/td>\n[latex]= 2[\/latex]<\/td>\n<\/tr>\n
     <\/td>\n <\/td>\nTotal:\u00a0<\/td>\n[latex]1982[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

     <\/p>\n

    As you can see, the number [latex]30412_5[\/latex] is equivalent to [latex]1982[\/latex] in base-[latex]10[\/latex]. We will say [latex]30412_5 = 1982_{10}[\/latex]. All of this may seem strange to you, but that\u2019s only because you are so used to the only system that you\u2019ve probably ever seen.<\/p>\n<\/section>\n

    \n

    How to: Convert from Base-[latex]b[\/latex] to Base-[latex]10[\/latex]<\/strong><\/p>\n

      \n
    1. Identify the place value for each digit in the base-[latex]b[\/latex] number, starting from the rightmost place as the “units” place (equivalent to base-[latex]b^0[\/latex]), and then increasing the power by one with each position to the left.<\/li>\n
    2. Multiply each digit of the base-[latex]b[\/latex] number by the corresponding base-[latex]b[\/latex] place value.<\/li>\n
    3. Sum up the values obtained in the second step, which results in the equivalent base-[latex]10[\/latex] number.<\/li>\n
    4. Proceed in this way, from right to left, until every digit of the base-[latex]b[\/latex] number has been processed.<\/li>\n
    5. The total sum after all operations is your converted base-[latex]10[\/latex] number.<\/li>\n<\/ol>\n<\/section>\n
      Convert [latex]6234_{7}[\/latex]\u00a0to a base-[latex]10[\/latex] number.<\/p>\n