## Scan and Solve with PhotoMath

PhotoMath is a new app which allows users to scan a math expressions or equations using a phone camera in real time and solve them. As of this writing, the supported expressions/equations are the arithmetic operations (+, -, ×,÷), fractions, decimals, powers and roots, linear equations, powers and roots, quadratic equations, simple systems of linear equations. It also supports absolute value equations and inequalities.

Below is a short introductory video about PhotoMath.

https://vimeo.com/109405701

PhotoMath is available as an iOS, Android, Amazon and Windows Phone application.

## Understanding the Concept of Inverse Functions

Let us consider the functions

$f(x) = x^3$  and $g(x) = \sqrt[3]{x}$.

The table on the left shows the ordered pairs $(x,f(x))$. We used $f(x)$ and substitute them to $x$ in the second table.

As we can see, all the values of $g(x)$ are the same as those of $x$ in the first table. For example,  if we have $x = 3$ in the first table, applying $f$, we get  27 as output. On the other hand, if we apply $g$ to 27, the value returns to $3$

## Why the Area of a Rhombus is Half the Product of its Diagonals

A rhombus is a parallelogram with four congruent sides. Since it is a parallelogram, it has also all the properties of a parallelogram. One of these properties is that the diagonals bisect each other. That is, they divide each other into two equal parts.

Another property of a rhombus is that the diagonals are perpendicular. So, summarizing all the properties above, if we have rhombus $ABCD$, then,

$\overline{AB} \cong \overline{BC} \cong \overline{CD} \cong \overline{DA}$.

and

$\overline{AC} \perp \overline{BD}$.