# freshman’s dream error

The name “freshman’s dream theorem^{}” comes from the fact that people who are unfamiliar with mathematics commonly make the error of distributing exponents over addition^{} and/or subtraction, typically when working in fields of characteristic zero. An example is the equation ${(x+y)}^{2}={x}^{2}+{y}^{2}$ for $x,y\in \mathbb{R}$. The equation is incorrect unless $x=0$ or $y=0$. By no means does the exponent need to be a natural number^{} or an integer for this error to occur. An example of this is the equation $\sqrt{x+y}=\sqrt{x}+\sqrt{y}$ for $x,y\in \mathbb{R}$ with $x\ge 0$ and $y\ge 0$. This equation can be rewritten using the exponent $\frac{1}{2}$, and again, the equation is incorrect unless $x=0$ or $y=0$.

An easy way to explain to someone who is under the impression that exponents distribute over addition and/or subtraction is to provide a counterexample. For instance, when $x=3$ and $y=4$, we have:

$\begin{array}{ccccccc}\hfill {(x+y)}^{2}\hfill & \hfill =\hfill & \hfill {(3+4)}^{2}\hfill & \hfill =\hfill & \hfill {7}^{2}\hfill & \hfill =\hfill & \hfill 49\hfill \\ & & & & & & \\ \hfill {x}^{2}+{y}^{2}\hfill & \hfill =\hfill & \hfill {3}^{2}+{4}^{2}\hfill & \hfill =\hfill & \hfill 9+16\hfill & \hfill =\hfill & \hfill 25\hfill \end{array}$

On the other hand, the freshman’s dream theorem yields some instances in which exponents can be distributed over addition and/or subtraction.

Title | freshman’s dream error |
---|---|

Canonical name | FreshmansDreamError |

Date of creation | 2013-03-22 16:07:23 |

Last modified on | 2013-03-22 16:07:23 |

Owner | Wkbj79 (1863) |

Last modified by | Wkbj79 (1863) |

Numerical id | 7 |

Author | Wkbj79 (1863) |

Entry type | Example |

Classification | msc 97D70 |