Example: Keywords and Modifiers Combination

Symbolics > Working with Symbolics > Example: Keywords and Modifiers Combination

Keyword assume with modifiers real and complex

1. Write and evaluate expressions using rectangular assuming x is real and complex.

<region id="ID0E4AWLB" actualWidth="285.72000000000008" actualHeight="39.2" top="172.8" left="96.000000000000014" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="14">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <id labels="FUNCTION" label-is-contextual="true" xml:space="preserve">exp</id>
        <id xml:space="preserve">x</id>
      </apply>
      <command>
        <sequence>
          <sequence>
            <id xml:space="preserve">rectangular</id>
            <id xml:space="preserve">assume</id>
          </sequence>
          <apply>
            <equal />
            <id xml:space="preserve">x</id>
            <id xml:space="preserve">real</id>
          </apply>
        </sequence>
      </command>
      <symResult>
        <apply>
          <pow />
          <id labels="CONSTANT" xml:space="preserve">e</id>
          <id labels="*" xml:space="preserve">x</id>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

<region id="ID0EPBWLB" actualWidth="434.27666666666676" actualHeight="39.2" top="230.40000000000003" left="96.000000000000014" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="15">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <id labels="FUNCTION" label-is-contextual="true" xml:space="preserve">exp</id>
        <id xml:space="preserve">x</id>
      </apply>
      <command>
        <sequence>
          <sequence>
            <id xml:space="preserve">rectangular</id>
            <id xml:space="preserve">assume</id>
          </sequence>
          <apply>
            <equal />
            <id xml:space="preserve">x</id>
            <id xml:space="preserve">complex</id>
          </apply>
        </sequence>
      </command>
      <symResult>
        <apply>
          <plus />
          <apply>
            <id labels="FUNCTION" xml:space="preserve">Re</id>
            <apply>
              <pow />
              <id labels="CONSTANT" xml:space="preserve">e</id>
              <id labels="*" xml:space="preserve">x</id>
            </apply>
          </apply>
          <apply>
            <mult />
            <apply>
              <id labels="FUNCTION" xml:space="preserve">Im</id>
              <apply>
                <pow />
                <id labels="CONSTANT" xml:space="preserve">e</id>
                <id labels="*" xml:space="preserve">x</id>
              </apply>
            </apply>
            <imag symbol="i">1</imag>
          </apply>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

The results are different based on assumption.

x is real by default

Keyword: simplify for values greater than 0

1. Write and simplify the expression assuming values less than or equal to 0 and greater than 0.

<region id="ID0EXEWLB" actualWidth="399.71400125185647" actualHeight="39.2" top="460.80000000000007" left="57.600000000000009" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="18">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <mult />
        <apply>
          <nthRoot />
          <placeholder />
          <apply>
            <minus />
            <id xml:space="preserve">a</id>
            <imag symbol="i">3</imag>
          </apply>
        </apply>
        <apply>
          <nthRoot />
          <placeholder />
          <apply>
            <plus />
            <id xml:space="preserve">a</id>
            <imag symbol="i">1</imag>
          </apply>
        </apply>
      </apply>
      <command>
        <sequence>
          <sequence>
            <id xml:space="preserve">simplify</id>
            <id xml:space="preserve">assume</id>
          </sequence>
          <apply>
            <lessOrEqual />
            <id xml:space="preserve">a</id>
            <real>0</real>
          </apply>
        </sequence>
      </command>
      <symResult>
        <apply>
          <mult />
          <apply>
            <nthRoot />
            <placeholder />
            <apply>
              <minus />
              <id labels="*" xml:space="preserve">a</id>
              <imag symbol="i">3</imag>
            </apply>
          </apply>
          <apply>
            <nthRoot />
            <placeholder />
            <apply>
              <plus />
              <id labels="*" xml:space="preserve">a</id>
              <imag symbol="i">1</imag>
            </apply>
          </apply>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

<region id="ID0EJFWLB" actualWidth="388.81133427222568" actualHeight="39.2" top="518.40000000000009" left="57.600000000000009" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="19">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <mult />
        <apply>
          <nthRoot />
          <placeholder />
          <apply>
            <minus />
            <id xml:space="preserve">a</id>
            <imag symbol="i">3</imag>
          </apply>
        </apply>
        <apply>
          <nthRoot />
          <placeholder />
          <apply>
            <plus />
            <id xml:space="preserve">a</id>
            <imag symbol="i">1</imag>
          </apply>
        </apply>
      </apply>
      <command>
        <sequence>
          <sequence>
            <id xml:space="preserve">simplify</id>
            <id xml:space="preserve">assume</id>
          </sequence>
          <apply>
            <greaterThan />
            <id xml:space="preserve">a</id>
            <real>0</real>
          </apply>
        </sequence>
      </command>
      <symResult>
        <apply>
          <nthRoot />
          <placeholder />
          <apply>
            <plus />
            <apply>
              <minus />
              <apply>
                <pow />
                <id labels="*" xml:space="preserve">a</id>
                <real>2</real>
              </apply>
              <apply>
                <mult />
                <imag symbol="i">2</imag>
                <id labels="*" xml:space="preserve">a</id>
              </apply>
            </apply>
            <real>3</real>
          </apply>
        </apply>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
    <formattingOverride BackgroundColor="#FFC3D9A5" />
  </math>
</region>

The results are different based on assumption.

Keyword: solve and modifier real

1. Write an equation and solve it.

<region id="ID0ETIWLB" actualWidth="510.74" actualHeight="100" top="729.60000000000014" left="19.200000000000003" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="20">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <plus />
        <apply>
          <minus />
          <apply>
            <pow />
            <id xml:space="preserve">x</id>
            <real>5</real>
          </apply>
          <apply>
            <pow />
            <id xml:space="preserve">x</id>
            <real>2</real>
          </apply>
        </apply>
        <real>1</real>
      </apply>
      <command>
        <id xml:space="preserve">solve</id>
      </command>
      <symResult>
        <matrix rows="5" cols="1">
          <real>-0.80873060047939201374</real>
          <apply>
            <plus />
            <real>-0.46491220160289785433</real>
            <imag symbol="i">1.0714738402702694092</imag>
          </apply>
          <apply>
            <minus />
            <real>-0.46491220160289785433</real>
            <imag symbol="i">1.0714738402702694092</imag>
          </apply>
          <apply>
            <minus />
            <real>0.8692775018425938612</real>
            <imag symbol="i">0.38826940659974035536</imag>
          </apply>
          <apply>
            <plus />
            <real>0.8692775018425938612</real>
            <imag symbol="i">0.38826940659974035536</imag>
          </apply>
        </matrix>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

2. Solve the equation again assuming x is real.

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  <math resultRef="21">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <plus />
        <apply>
          <minus />
          <apply>
            <pow />
            <id xml:space="preserve">x</id>
            <real>5</real>
          </apply>
          <apply>
            <pow />
            <id xml:space="preserve">x</id>
            <real>2</real>
          </apply>
        </apply>
        <real>1</real>
      </apply>
      <command>
        <sequence>
          <sequence>
            <id xml:space="preserve">solve</id>
            <id xml:space="preserve">assume</id>
          </sequence>
          <apply>
            <equal />
            <id xml:space="preserve">x</id>
            <id xml:space="preserve">real</id>
          </apply>
        </sequence>
      </command>
      <symResult>
        <real>-0.80873060047939201374</real>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

The result is filtered and displays only real values of the solution.

Keyword assume and modifiers even and odd

1. Write an expression and solve it using solve.

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  <math resultRef="22">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <plus />
        <apply>
          <minus />
          <apply>
            <minus />
            <apply>
              <pow />
              <id xml:space="preserve">x</id>
              <real>5</real>
            </apply>
            <apply>
              <mult />
              <real>4</real>
              <apply>
                <pow />
                <id xml:space="preserve">x</id>
                <real>3</real>
              </apply>
            </apply>
          </apply>
          <apply>
            <pow />
            <id xml:space="preserve">x</id>
            <real>2</real>
          </apply>
        </apply>
        <real>4</real>
      </apply>
      <command>
        <id xml:space="preserve">solve</id>
      </command>
      <symResult>
        <matrix rows="5" cols="1">
          <real>2</real>
          <real>1</real>
          <real>-2</real>
          <apply>
            <div />
            <apply>
              <minus />
              <apply>
                <mult />
                <apply>
                  <nthRoot />
                  <placeholder />
                  <real>3</real>
                </apply>
                <imag symbol="i">1</imag>
              </apply>
              <real>1</real>
            </apply>
            <real>2</real>
          </apply>
          <apply>
            <div />
            <apply>
              <minus />
              <apply>
                <neg />
                <parens>
                  <apply>
                    <mult />
                    <apply>
                      <nthRoot />
                      <placeholder />
                      <real>3</real>
                    </apply>
                    <imag symbol="i">1</imag>
                  </apply>
                </parens>
              </apply>
              <real>1</real>
            </apply>
            <real>2</real>
          </apply>
        </matrix>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

2. Evaluate the equation assuming x is integer.

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  <math resultRef="23">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <plus />
        <apply>
          <minus />
          <apply>
            <minus />
            <apply>
              <pow />
              <id xml:space="preserve">x</id>
              <real>5</real>
            </apply>
            <apply>
              <mult />
              <real>4</real>
              <apply>
                <pow />
                <id xml:space="preserve">x</id>
                <real>3</real>
              </apply>
            </apply>
          </apply>
          <apply>
            <pow />
            <id xml:space="preserve">x</id>
            <real>2</real>
          </apply>
        </apply>
        <real>4</real>
      </apply>
      <command>
        <sequence>
          <sequence>
            <id xml:space="preserve">solve</id>
            <id xml:space="preserve">assume</id>
          </sequence>
          <apply>
            <equal />
            <id xml:space="preserve">x</id>
            <id xml:space="preserve">integer</id>
          </apply>
        </sequence>
      </command>
      <symResult>
        <matrix rows="3" cols="1">
          <real>2</real>
          <real>1</real>
          <real>-2</real>
        </matrix>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

3. Evaluate the equation assuming x is even.

<region id="ID0EAQWLB" actualWidth="335.08" actualHeight="49.49" top="1209.6000000000001" left="76.800000000000011" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="24">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <plus />
        <apply>
          <minus />
          <apply>
            <minus />
            <apply>
              <pow />
              <id xml:space="preserve">x</id>
              <real>5</real>
            </apply>
            <apply>
              <mult />
              <real>4</real>
              <apply>
                <pow />
                <id xml:space="preserve">x</id>
                <real>3</real>
              </apply>
            </apply>
          </apply>
          <apply>
            <pow />
            <id xml:space="preserve">x</id>
            <real>2</real>
          </apply>
        </apply>
        <real>4</real>
      </apply>
      <command>
        <sequence>
          <sequence>
            <id xml:space="preserve">solve</id>
            <id xml:space="preserve">assume</id>
          </sequence>
          <apply>
            <equal />
            <id xml:space="preserve">x</id>
            <id xml:space="preserve">even</id>
          </apply>
        </sequence>
      </command>
      <symResult>
        <matrix rows="2" cols="1">
          <real>2</real>
          <real>-2</real>
        </matrix>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

4. Evaluate the equation assuming x is odd.

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  <math resultRef="25">
    <symEval xmlns="http://schemas.mathsoft.com/math50">
      <apply>
        <plus />
        <apply>
          <minus />
          <apply>
            <minus />
            <apply>
              <pow />
              <id xml:space="preserve">x</id>
              <real>5</real>
            </apply>
            <apply>
              <mult />
              <real>4</real>
              <apply>
                <pow />
                <id xml:space="preserve">x</id>
                <real>3</real>
              </apply>
            </apply>
          </apply>
          <apply>
            <pow />
            <id xml:space="preserve">x</id>
            <real>2</real>
          </apply>
        </apply>
        <real>4</real>
      </apply>
      <command>
        <sequence>
          <sequence>
            <id xml:space="preserve">solve</id>
            <id xml:space="preserve">assume</id>
          </sequence>
          <apply>
            <equal />
            <id xml:space="preserve">x</id>
            <id xml:space="preserve">odd</id>
          </apply>
        </sequence>
      </command>
      <symResult>
        <real>1</real>
      </symResult>
    </symEval>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

The results vary based on assumption. Observe that result for even and odd combined result in the integer.

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