Let us recall the following notes.
- Procedure to write the product of a number with \(10000...1\) in a single step:
In general, the method to find the product when one of the factors of the product is of the form \(1000...1 = 10^n+1\), while the other factor is any number with an arbitrary number of digits is as follows:
Step -1: Begin with the last digits of the number. Write it down the last \(n\) digits as it is.
Step - 2: From right to left, add each number with the \(n\)th next number and write the sum in between. If the sum is \(\geq 10\), carry over the tens digit to the next place.
Step - 3: Finally, write down the initial \(n\) digits of the number, with any carry over if applicable.
Step - 2: From right to left, add each number with the \(n\)th next number and write the sum in between. If the sum is \(\geq 10\), carry over the tens digit to the next place.
Step - 3: Finally, write down the initial \(n\) digits of the number, with any carry over if applicable.
- Algebraic Identities
General expression for the square of sum of two numbers:
Identity 1A: \((a+b)^{2}\) \(=\) \(a^{2} + 2ab + b^{2}\)
General expression for the square of difference of two numbers:
Identity 1B: \((a-b)^{2}\) \(=\) \(a^{2} - 2ab + b^{2}\)
General expression for \((a+b)(a-b)\):
Identity 1C: \((a+b)(a-b)\) \(=\) \(a^2 - b^2\)
- Investigating a pattern:
To investigate a pattern means looking for how something changes or grows and trying to find a rule or formula that describes it.
This process involves the following:
This process involves the following:
1: Observing the arrangement or sequence carefully.
2: Recording your observations, the first few terms, differences, or shapes that you see.
2: Recording your observations, the first few terms, differences, or shapes that you see.
3: Predicting what might come next.
4: Formulating a general rule using variables to describe the pattern.
5: Testing the general rule with new terms to see if it always works.
4: Formulating a general rule using variables to describe the pattern.
5: Testing the general rule with new terms to see if it always works.
There are often several ways to observe and interpret a pattern in mathematics. Approaching a problem with different perspective encourages creative and imaginative thinking.
- Multiple methods to solve a problem
In mathematics, diagrams and figures often permit multiple interpretations of the same region or shape. This means that a single problem can be solved through multiple valid methods. Such an approach helps us derive a relation between geometric shapes and algebraic expressions.
The concept of "multiple methods" involves the ability to interpret one diagram in two or more ways, yet arrive at the same area or value using different strategies.
Important!
- A single geometric figure can be interpreted in more than one way, and each method produces the same result.
- This approach helps us derive and understand algebraic expressions and identities geometrically.