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.
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.
Here is my youtube video on this topic :
Try out the questions below based on this topic :
