Example: Finding Digits and Improving Result Precision

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Example: Finding Digits and Improving Result Precision

To find the digit occupying a particular place within a number, divide the number by its base raised to the desired decimal place minus 1, and find the floor of the remainder.

1. Define a decimal number and the place of the desired digit within it.

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Click to copy this expression

2. Use functions floor and mod to find the digit in the 5th place.

Click to copy this expression

<region id="ID0ESRABB" top="307.20000000000005" left="38.400000000000006" actualWidth="197.59333333333333" actualHeight="49.352000000000004" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="7">
    <eval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">floor</id>
        <apply>
          <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">mod</id>
          <sequence>
            <apply>
              <div />
              <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">x</id>
              <apply>
                <pow />
                <real>10</real>
                <apply>
                  <minus />
                  <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">n</id>
                  <real>1</real>
                </apply>
              </apply>
            </apply>
            <real>10</real>
          </sequence>
        </apply>
      </apply>
      <unitOverride>
        <placeholder />
      </unitOverride>
    </eval>
    <resultFormat>
      <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

3. Show that the order of calculations can affect the precision of the results.

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<region id="ID0EDTABB" top="441.60000000000008" left="38.400000000000006" actualWidth="309.35666666666668" actualHeight="47.2" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="8">
    <eval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <minus />
        <real>6000</real>
        <apply>
          <mult />
          <apply>
            <mult />
            <real>120</real>
            <parens>
              <apply>
                <div />
                <real>200</real>
                <id labels="CONSTANT" xml:space="preserve" label-is-contextual="true">π</id>
              </apply>
            </parens>
          </apply>
          <apply>
            <div />
            <id labels="CONSTANT" xml:space="preserve" label-is-contextual="true">π</id>
            <real>4</real>
          </apply>
        </apply>
      </apply>
      <unitOverride>
        <placeholder />
      </unitOverride>
    </eval>
    <resultFormat>
      <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

Click to copy this expression

<region id="ID0ETTABB" top="499.20000000000005" left="38.400000000000006" actualWidth="197.62" actualHeight="47.2" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="9">
    <eval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <minus />
        <real>6000</real>
        <apply>
          <mult />
          <real>120</real>
          <parens>
            <apply>
              <mult />
              <parens>
                <apply>
                  <div />
                  <real>200</real>
                  <id labels="CONSTANT" xml:space="preserve" label-is-contextual="true">π</id>
                </apply>
              </parens>
              <apply>
                <div />
                <id labels="CONSTANT" xml:space="preserve" label-is-contextual="true">π</id>
                <real>4</real>
              </apply>
            </apply>
          </parens>
        </apply>
      </apply>
      <unitOverride>
        <placeholder />
      </unitOverride>
    </eval>
    <resultFormat>
      <decimal precision="round-trip-decimal-places" show-trailing-zeros="false" radix="dec" imaginary-value="i" />
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

In the second case, there is no roundoff error because the expression inside the outside bracket is evaluated first without converting π to a number.

Therefore, you can improve the precision of your results by minding the order of operations.