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 \(1\).
Let us find the increase in product for \(a (b +1)\).
Expand the product using distributive property \(a(b + c) = ab + ac\).
\(a (b + 1) = ab + \fbox{a}\)
Observe that the product \(ab\) is increased by \(a\).
Example:
Consider the product of \(a = 21\) and \(b = 19\).
Let us find the change in product if \(19\) is increased by \(1\).
\(21 (19 + 1) = 21 \times 19 + 21 \times 1\)
\(=\) \(21 \times 19\) \(+\) \(\fbox{21}\)
The product is increased by \(21\).
Let us find the increase in product for \((a + 1)b\).
Expand the product using distributive property \((a + b)c = ac + bc\).
\((a + 1)b = ab + \fbox{b}\)
Observe that the product \(ab\) is increased by \(b\).
Example:
Consider the product of \(a = 21\) and \(b = 19\)
Let us find the change in product if \(21\) is increased by \(1\).
\((21 + 1)19 = 21 \times 19 + 1 \times 19\)
\(=\) \(21 \times 19\) \(+\) \(\fbox{19}\)
The product is increased by \(19\).
Let us find the increase in product for \((a + 1)(b +1)\).
Expand the product using the distributive property of multiplication.
\((a +1)(b + 1) = (a + 1)b + (a + 1)1\)
\(=\) \(ab + \fbox{b + a + 1}\)
Observe that the product \(ab\) is increased by \(b + a + 1\).
Example:
Consider the product of \(a = 21\) and \(b = 19\).
Let us find the change in product if both \(21\) and \(19\) are increased by \(1\).
\((21 +1) (19 + 1) = (21 +1) 19 + (21 +1)1\)
\(=\) \(21 \times 19 + 19 + 21 + 1\)
\(=\) \(21 \times 19\) \(+\) \(\fbox{19 + 21 + 1}\)
The product is increased by \(19 + 21 + 1\).
Let us find the increase in product for \((a + 1)(b -1)\).
Expand the product using the distributive property of multiplication.
\((a + 1)(b - 1) = (a + 1)b - (a + 1)1\)
\(=\) \(ab + \fbox{b - a - 1}\)
Observe that the product \(ab\) is increased by \(b - a - 1\).
Example:
Consider the product of \(a = 19\) and \(b = 23\).
Let us find the change in product if \(19\) is increased by \(1\) and \(23\) is decreased by \(1\).
\((19 +1) (23 - 1) = (19 +1) 23 - (19 +1)1\)
\(=\) \(19 \times 23 + 23 - 19 - 1\)
\(=\) \(19 \times 23\) \(+\) \(\fbox{23 - 19 - 1}\)
The product is increased by \(23 - 19 - 1\).
Important!
The increment in product does not always imply that the product will always increase. The product may increase or decrease based on the values of \(a\) and \(b\), depending on the type of integer choosen.
The way of generalising any algebraic expression is a method to derive an identity.
Mathematical statements that express the equality of two algebraic expression are called identities.