The distributive property is one of the most frequently used properties in mathematics. This property defines the relation between multiplication and addition, and is extensively used in arithmetic as well as algebraic simplifications.
The property states that when an integer or an expression is multiplied by a sum, the factor is multiplied to each term separately, and then the products are added.
For any three real numbers \(a\), \(b\) and \(c\)
- \(a(b +c) = ab + ac\)
The product of any two real numbers is represented by \(a \times b\).
In this chapter we will analyse the increase in product if either \(a\) or \(b\) or both \(a\) and \(b\) are increased by \(1\).
Case (i): In the product \(a \times b\), the second number \(b\) is increased by \(1\), then the product \(ab\) is increased by \(a\).
\(a (b + 1) = ab + \fbox{a}\)
Case (ii): In the product \(a \times b\), the first number \(a\) is increased by \(1\), then the product \(ab\) is increased by \(b\).
\((a + 1)b = ab + \fbox{b}\)
Case (iii): In the product \(a \times b\), both \(a\) and \(b\) are increased by \(1\), then the product \(ab\) is increased by \(b + a + 1\).
\((a +1)(b + 1) = ab + \fbox{b + a + 1}\)
Case (iv): In the product \(a \times b\), if \(a\) is increased by \(1\) and \(b\) is decreased by \(1\), then the product \(ab\) is increased by \(b - a - 1\).
\((a + 1)(b - 1) = ab + \fbox{b - a - 1}\)
Case (v): In the product \(a \times b\), if \(a\) is decreased by \(1\) and \(b\) is increased by \(1\), then the product \(ab\) is increased by \(a - b - 1\).
\((a - 1)(b + 1) = ab + \fbox{- b + a - 1}\)
Case (vi): In the product \(a \times b\), both \(a\) and \(b\) are decreased by \(1\), then the product \(ab\) is increased by \(- b - a + 1\).
\((a -1)(b - 1) = ab + \fbox{- b - a + 1}\)
Now, we will generalise the increase or decrease in the product \(ab\), if both \(a\) and \(b\) are increased or decreased by two different numbers say \(m\) and \(n\).
In the product \(ab\), the first number \(a\) is increased by a number \(m\) and the second number \(b\) is increased by a number \(n\).
- \((a + m)(b + n) = ab + mb + an + mn\)
- \((a + u)(b - v) = ab + ub - av - uv\)
- \((a - u)(b + v) = ab - ub + av - uv\)
- \((a - u)(b - v) = ab - ub - av + uv\)