IDENTITIES

 IDENTITIES

LEARNING OUTCOMES

From this you will understand:

• the concept “Identities”.
• the geometric expression of product of two numbers.
• the geometric expression of square of a number.
• the method of multiplication between two large numbers by splitting each into two positive numbers.
• the method of multiplication between two sums of positive numbers.

DETAILED NOTES:

IDENTITIES

         Those equations which are true for all numbers are called Identities.

Example :

         x+(x+1) = 2x+1.

Here,

when x=1,  x+(x+1) = 1+(1+1)= 3    

  2x+1 = 2×1 + 1 = 3

when x=2,  x+(x+1) = 2+(2+1) = 5

  2x+1 = 2×2 + 1 = 5

That is, both sides gets equal if we give any number instead of x. You can try with other numbers also.  

But 3x+2=2x+3 is not an identity. Because,

When x = 1, 3x+2 = 3×1 + 2 = 5

   2x+3 = 2×1 + 3 = 5

When x = 2, 3x+2 = 3×2 + 2 = 8

   2x+3 = 2×2 + 3 = 7 

That is, both sides gets equal when x=1, but that doesn't happen when x=2.  Hence it is not an identity.

Have a look at some other important keynotes :

• The product of two numbers is same as the area of a rectangle with sides those two numbers.

         Example : 

The product 8×3 can be expressed as the area of a rectangle with sides 8 cm and 3 cm.

• The square of a number is same as the area of a square with side that given number.

Example:

The square of the number 5, 5×5  can be expressed as the area of a square with side 5 cm.

 

PRODUCT OF SUMS

To multiply a sum of positive numbers by a sum of positive numbers, multiply each number in the second sum by each number in the first sum and add.

That is, for any positive numbers x,y,u,v,

         (x+y)(u+v) = xu + xv + yu + yv

Example :

     35 × 24 = (30+5)(20+4)

                   = (30×20) + (30×4) + (5×20) + (5×4)

                   = 600 + 120 + 100 + 20

                   = 840.

     Try out this : 

     105 × 92 

      ( Hint : Split 105 = 100 + 5 , 92 = 90+2 )

Here is the geometrical expression for the product of sums of two numbers.

Suppose we have a rectangle with sides 'x' units and 'u' units. If we increase each sides by 'y' units and 'v' units respectively, and complete it as a rectangle we get the area of the larger rectangle with sides x+y and u+v as the sum of areas of 4 smaller rectangles.

Can this concept be connected to something that we use in our daily life?

If I say the answer as 'Calendar',you probably think it as a funny joke. But it's a fact that you can use the concept discussed above in calender math and there are some problems also using this idea.

Here is a video of Calendar math and some problems associated with it.

Its a must watch one. 

Summary : 

• Those equations which are true for all numbers are called identities.
• To multiply a sum of positive numbers by a sum of positive numbers, multiply each number in the second sum by each number in the first sum and add.
• (x+y)(u+v) = xu + xv + yu + yv.

Click here to view my PPT

Here is my youtube video on this topic :

Try out the questions below based on this topic :