*The study of non-Euclidean Geometry brings nothing to students but fatigue, vanity, arrogance, and imbecility. Non-Euclidean space is the false invention of demons, who gladly furnish the dark understanding of the non-Euclideans with false knowledge. The non-Euclideans, like the ancient sophists, seem unaware that their understandings have become obscured by the promptings of the evil spirits.*

**Matthew RYAN**

Welcome to the blog **Math1089 – Mathematics for All**.

Glad you came by. I wanted to let you know I appreciate your spending time here at the blog very much. I do appreciate your taking time out of your busy schedule to check out **Math1089**!

Squaring a number means multiplying that number by itself. For example, 3 × 3 = 9 is the square of 3; 11 × 11 = 121 is the square of 11 etc. This method is available for finding squares in conventional mathematics. But in Vedic mathematics, there are smarter ways available to find the square of specific numbers.

In this blog post, we are going to discuss about the *shortcut rule* for finding the square of numbers, **ending in 5**. The trick is very much well-known in Vedic mathematics and useful for various competitive examinations. The rule is commonly known as *by one more than the one before*.

The rule is given below:

∎ (**a**) Divide the given number in **two parts**: (**A**) one part is the *single-digit number* (5) and (**B**) the other part is the number formed by the *rest of the digits* (this number can be of one, two, three, . . . digit/s).

∎ (**b**) The answer comes in two parts – **RHS** and **LHS** and the *final answer* looks like (**LHS**)(**RHS**), where

(i) **RHS** is always 5^{2} (= 25), as the number ends in 5;

(ii) **LHS** is computed by *multiplying the number formed by the digits before* 5 *in the given number by its successor*. In other words, **we multiply the number obtained in (B) by its next integer**.

**Example 1**. Find the square of 75.

*Solution*. Let us divide the *given number* 75 in two parts like

(**A**) 5 and (**B**) 7 (a single-digit number and its successor is 8).

Then we have

(i)

RHSof 75^{2}= 5^{2 }= 25;

(ii)

LHSof 75^{2}= (

the number formed by the digits before5in the given number) × (its successor)= 7 × 8

= 56.

The final answer is (**LHS**)(**RHS**) = 5625.

**Example 2**. Find the square of 345.

*Solution*. Let us divide the *given number* 345 in two parts like

(**A**) 5 and (**B**) 34 (a two-digit number and its successor is 35).

Then we have

(i)

RHSof 345^{2}= 5^{2 }= 25;

(ii)

LHSof 345^{2}= (

the number formed by the digits before5in the given number) × (its successor)= 34 × 35

= 1190.

The final answer is (**LHS**)(**RHS**) = 119025.

**Example 3**. Find the square of 8005.

*Solution*. Let us divide the *given number* 8005 in two parts like

(**A**) 5 and (**B**) 800 (a three-digit number and its successor is 801).

Then we have

(i)

RHSof 8005^{2}= 5^{2 }= 25;

(ii)

LHSof 8005^{2}= (

the number formed by the digits before5in the given number) × (its successor)= 800 × 801

= 640800.

The final answer is (**LHS**)(**RHS**) = 64080025.

Find the square of the following numbers:

- (
**a**) 15 - (
**b**) 85 - (
**c**) 985 - (
**d**) 215 - (
**e**) 1015 - (
**f**) 2075

Your suggestions are eagerly and respectfully welcome! See you soon with a new mathematics blog that you and I call **“****Math1089 – Mathematics for All!**“.