Squares and Square Roots

1 Squares AND Square Roots IntroductionYou know that the area of a Square = side side (where side means the length ofa side ). Study the following of a Square (in cm)Area of the Square (in cm2)11 1 = 1 = 1222 2 = 4 = 2233 3 = 9 = 3255 5 = 25 = 5288 8 = 64 = 82aa a = a2 What is special about the numbers 4, 9, 25, 64 and other such numbers?Since, 4 can be expressed as 2 2 = 22, 9 can be expressed as 3 3 = 32, all suchnumbers can be expressed as the product of the number with numbers like 1, 4, 9, 16, 25, .. are known as Square general, if a natural number m can be expressed as n2, where n is also a naturalnumber, then m is a Square number.

2 Is 32 a Square number?We know that 52 = 25 and 62 = 36. If 32 is a Square number, it must be the Square ofa natural number between 5 and 6. But there is no natural number between 5 and 32 is not a Square the following numbers and their 1 = 122 2 = 4 Squares and SquareRootsCHAPTER62021 2290 MATHEMATICSTRY THESE33 3 = 944 4 = 1655 5 = 256-----------7-----------8-----------9- ----------10-----------From the above table, can we enlist the Square numbers between 1 and 100? Arethere any natural Square numbers upto 100 left out?You will find that the rest of the numbers are not Square numbers 1, 4, 9, 16.

3 Are Square numbers. These numbers are also called the perfect Square numbers between (i) 30 and 40 (ii) 50 and Properties of Square NumbersFollowing table shows the Squares of numbers from 1 to Square1111121241214439131694161419652515 2256361625674917289864183249811936110100 20400 Study the Square numbers in the above table. What are the ending digits (that is, digits inthe units place) of the Square numbers? All these numbers end with 0, 1, 4, 5, 6 or 9 atunits place. None of these end with 2, 3, 7 or 8 at unit s we say that if a number ends in 0, 1, 4, 5, 6 or 9, then it must be a squarenumber?

4 Think about we say whether the following numbers are perfect Squares ? How do we know?(i)1057(ii)23453(iii)7928(iv)222222 (v)1069(vi)2061 TRY THESECan youcomplete it?2021 22 Squares AND Square Roots 91 Write five numbers which you can decide by looking at their units digit that they arenot Square five numbers which you cannot decide just by looking at their units digit(or units place) whether they are Square numbers or not. Study the following table of some numbers and their Squares and observe the one splace in 1 NumberSquareNumberSquareNumberSquare1111 1212144124121442248439131692352941614196 2457652515225256256361625630900749172893 5122586418324401600981193614520251010020 400502500 The following Square numbers end with digit the next two Square numbers which end in 1 and their corresponding will see that if a number has 1 or 9 in the units place, then it s Square ends in 1.

5 Let us consider Square numbers ending in THESEW hich of 1232, 772, 822,1612, 1092 would end withdigit 1?TRY THESEW hich of the following numbers would have digit6 at unit place.(i)192(ii)242(iii)262(iv)362(v)342 2021 2292 MATHEMATICSTRY THESETRY THESEWe can see that when a Square number ends in 6, the number whose Square it is, willhave either 4 or 6 in unit s you find more such rules by observing the numbers and their Squares (Table 1)?What will be the one s digit in the Square of the following numbers?(i)1234(ii)26387(iii)52698(iv)99 880(v)21222(vi)9106 Consider the following numbers and their = 100202 =400802 =64001002 =100002002 =400007002 =4900009002 =810000If a number contains 3 zeros at the end, how many zeros will its Square have ?

6 What do you notice about the number of zeros at the end of the number and thenumber of zeros at the end of its Square ?Can we say that Square numbers can only have even number of zeros at the end? See Table 1 with numbers and their can you say about the Squares of even numbers and Squares of odd numbers? Square of which of the following numbers would be an odd number/an evennumber? Why?(i)727(ii)158(iii)269(iv) will be the number of zeros in the Square of the following numbers?(i)60(ii) Some More Interesting triangular you remember triangular numbers (numbers whose dot patterns can be arrangedas triangles)?

7 ** ** ** ** **1361015 But we havefour zerosBut we havetwo zerosWe haveone zeroWe havetwo zeros2021 22 Squares AND Square Roots 93If we combine two consecutive triangular numbers, we get a Square number, like1 + 3 = 43 + 6 = 96 + 10 = 16 = 22= 32= between Square numbersLet us now see if we can find some interesting pattern between two consecutivesquare (=12)2, 3, 4 (=22)5, 6, 7, 8, 9 (=32)10, 11, 12, 13, 14, 15, 16 (=42)17, 18, 19, 20, 21, 22, 23, 24, 25 (=52)Between 12(=1) and 22(= 4) there are two ( , 2 1) non Square numbers 2, 22(= 4) and 32(= 9) there are four ( , 2 2) non Square numbers 5, 6, 7, ,32 = 9, 42 = 16 Therefore,42 32 = 16 9 = 7 Between 9(=32) and 16(= 42) the numbers are 10, 11, 12, 13, 14, 15 that is, sixnon- Square numbers which is 1 less than the difference of two have42 = 16 and 52 = 25 Therefore,52 42 = 9 Between 16(= 42) and 25(= 52) the numbers are 17, 18.

8 , 24 that is, eight non squarenumbers which is 1 less than the difference of two 72 and 62. Can you say how many numbers are there between 62 and 72?If we think of any natural number n and (n + 1), then,(n + 1)2 n2 = (n2 + 2n + 1) n2 = 2n + find that between n2 and (n + 1)2 there are 2n numbers which is 1 less than thedifference of two , in general we can say that there are 2n non perfect Square numbers betweenthe Squares of the numbers n and (n + 1). Check for n = 5, n = 6 etc., and non Square numbersbetween the two squarenumbers 1 (=12) and 4(=22).

9 4 non Square numbersbetween the two squarenumbers 4(=22) and 9(32).8 non squarenumbers betweenthe two squarenumbers 16(= 42)and 25(=52).6 non Square numbers betweenthe two Square numbers 9(=32)and 16(= 42).2021 2294 MATHEMATICSTRY many natural numbers lie between 92 and 102 ? Between 112 and 122? many non Square numbers lie between the following pairs of numbers(i)1002 and 1012(ii)902 and 912(iii)10002 and odd numbersConsider the following1 [one odd number]= 1 = 121 + 3 [sum of first two odd numbers]= 4 = 221 + 3 + 5 [sum of first three odd numbers]= 9 = 321 + 3 + 5 + 7 [.]

10 ]= 16 = 421 + 3 + 5 + 7 + 9 [.. ]= 25 = 521 + 3 + 5 + 7 + 9 + 11 [.. ]= 36 = 62So we can say that the sum of first n odd natural numbers is at it in a different way, we can say: If the number is a Square number, it hasto be the sum of successive odd numbers starting from those numbers which are not perfect Squares , say 2, 3, 5, 6, .. Can youexpress these numbers as a sum of successive odd natural numbers beginning from 1?You will find that these numbers cannot be expressed in this the number 25. Successively subtract 1, 3, 5, 7, 9.