Key concept: Area of a square is side × side, i.e. if the side length is n, then area = n^2. In general, a natural number m is a square number if there exists a natural number n such that m = n^2.
Example table (conceptual): Side of square (cm) and Area (cm^2) follow A = s^2. For instance, 1^2 = 1, 2^2 = 4, 3^2 = 9, 5^2 = 25, 8^2 = 64, etc.
Notation: a × a = a^2. So square numbers are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, …
Is 32 a square number? Since 5^2 = 25 < 32 < 6^2 = 36 and there is no natural number between 5 and 6, 32 is not a square number.
Quick takeaways: square numbers are perfect squares of natural numbers; they are generated by squaring integers.
5.2 Properties of Square Numbers
A table of squares from 1 to 20 is given to study patterns (1^2, 2^2, 3^2, …, 20^2).
Ending digits of square numbers: the unit’s digit of a square is always one of ∈ {0, 1, 4, 5, 6, 9}. No square ends with 2, 3, 7, or 8 at the unit’s place.
Caution: The converse is not true. If a number ends with 0, 1, 4, 5, 6, or 9, it is not guaranteed to be a square (e.g., 1057 ends with 7? actually ends with 7; use examples from the exercise to test). The point is that a unit digit pattern alone cannot guarantee a perfect square.
Practice questions (examples): determine whether given numbers are perfect squares by comparing with known squares or by checking unit digits (i) 1057 (ii) 23453 (iii) 7928 (iv) 222222 (v) 1069 (vi) 2061.
Find perfect square numbers between intervals: (i) between 30 and 40 -> 36 is the only one; (ii) between 50 and 60 -> none (since 49 < 50 and 64 > 60).
5.3 Some More Interesting Patterns
Triangular numbers: a triangular number can be arranged as a triangle of dots. Examples: 1, 3, 6, 10, 15, …
Pattern: the sum of two consecutive triangular numbers gives a square: 1 + 3 = 4, 3 + 6 = 9, 6 + 10 = 16, i.e. sums of consecutive triangular numbers can yield squares: 1 + 3 = 2^2, 3 + 6 = 3^2, 6 + 10 = 4^2, etc.
Between consecutive squares, there are non-square integers: between n^2 and (n+1)^2 there are 2n numbers that are not perfect squares.
Derivation: (n+1)^2 − n^2 = 2n + 1, so there are (2n + 1) integers strictly between, hence 2n non-square numbers between n^2 and (n+1)^2.
Examples: between 1^2 = 1 and 2^2 = 4 there are 2 non-squares: 2, 3; between 4 and 9 there are 4 non-squares: 5, 6, 7, 8; between 9 and 16 there are 6 non-squares: 10–15; etc.
Observations: difference of consecutive squares gives 2n + 1; this explains patterns in spacing between squares.
5.4 Finding the Square of a Number
5.4.1 Quick patterns for squares
If a number ends in 5, its square has a special form. Let the number be of the form 10a+5. Then (10a+5)2=100a(a+1)+25.
This explains the last two digits are always 25 and the preceding digits are given by the product a(a+1).
Examples: 252=625 (a = 2 gives 2×3 = 6 in the hundreds place), 752=5625, 1252=15625, etc.
5.4.2 Pythagorean triplets
A Pythagorean triplet is a set of three positive integers (a, b, c) such that a2+b2=c2.
A useful form: for any natural number m > 1, (2m)2+(m2−1)2=(m2+1)2, giving the triplet (2m,m2−1,m2+1)
Examples:
m = 3: (6, 8, 10) since 62+82=36+64=100=102
m = 4: (8, 15, 17) since 82+152=64+225=289=172
Another well-known triplet: (5, 12, 13).
Note: Not all Pythagorean triples arise from the (2m, m^2-1, m^2+1) pattern; the general form is (a,b,c)=(m2−n2,2mn,m2+n2) with integers m > n.
5.5 Square Roots
Definition: The square root is the inverse operation of squaring. The positive square root of a natural number N is denoted by <br/>∇N or more standardly by \
oot N on? ; in common notation, extsqrt(N) or N is used for the non-negative root. The text emphasizes the positive root: for example, 4=2 (not -2).
5.5.1 Finding square roots (brief overview)
A few quick examples: 12=1⇒1=1, 22=4⇒4=2, 32=9⇒9=3.
5.5.2 Finding square root by repeated subtraction (sum of odd numbers)
The sum of the first n odd natural numbers equals n^2: 1+3+5+⋯+(2n−1)=n2.
To test if a number is a perfect square using this idea, repeatedly subtract consecutive odd numbers starting from 1 and observe whether you reach 0 exactly; if you do, the original number is a perfect square with root equal to the count of subtractions.
5.5.3 Finding square root through prime factorisation
If the prime factorisation of N is N=p<em>1e</em>1p<em>2e</em>2⋯, then for N to be a perfect square, all exponents must be even. In the square of a number, each prime factor exponent doubles: if a number has factorization K=∏p<em>if</em>i, then K2=∏p<em>i2f</em>i.
To find the square root via factorisation, pair primes: e.g., 324 = 2^2 × 3^4 = (2 × 3 × 3)^2 = 18^2, so sqrt(324) = 18.
Examples worked: sqrt(256) = 16; for 90 (2 × 3^2 × 5), not a square because 2 and 5 are unpaired; to make a square, multiply by 2 × 5 = 10 to get 900 with sqrt 30, etc.
5.5.4 Finding square root by division method (long division method)
Useful for large numbers; involves placing bars over pairs of digits from the decimal point outward, much like long division.
Steps (high-level):
1) Pair digits from the left of the decimal; determine the leftmost divisor by finding the largest square less than or equal to the leftmost group.
2) Subtract, bring down the next pair, and form a new divisor by doubling the current root and placing a blank digit to the right; choose the largest digit that keeps the product not exceeding the current dividend.
3) Repeat until all digits are exhausted. If decimals are present, process the decimal part with the decimal point in the quotient.
Example walkthroughs included: sqrt(529) = 23 and sqrt(4096) = 64; also demonstrates decimal square roots like sqrt(17.64) = 4.2.
5.6 Square Roots of Decimals
Method: apply the same “bars” method to both the integer and decimal parts, placing a decimal point in the quotient when the decimal part is encountered.
Example: sqrt(17.64) = 4.2; sqrt(12.25) = 3.5.
Note on digit counts: placing bars on decimal numbers helps determine the number of digits in the square root of a perfect square, and the number of digits in the root is related to the number of digit-pairs (bars) in the number.
5.7 What Have We Discussed? (Summary of key ideas)
A natural number m is a square number iff m = n^2 for some natural n.
Square numbers end in digits from the set {0, 1, 4, 5, 6, 9} in base 10.
A square number can have zeros at the end only in even counts (i.e., even number of trailing zeros).
The square root is the inverse of squaring; every perfect square has two integral square roots, +√N and −√N, but we usually consider the positive root when speaking of square roots.
Denotation: the positive square root of N is written as N; for example, 64=8.
Key formulas and results to remember
Area of a square: A=s2 where s is the side length.
A number is a square iff it can be written as n2 for some integer n: m=n2(square number).
End-digits of squares: Units digit∈0,1,4,5,6,9 for all squares.
Between consecutive squares: there are 2n non-square integers between n2 and (n+1)2 because there are 2n+1 integers in total between, minus the two endpoints.
Sums of odd numbers: 1+3+5+⋯+(2n−1)=n2; equivalently, a square number can be expressed as a sum of consecutive odd natural numbers starting from 1.
Pattern for numbers ending in 5: For a number of the form 10a+5, its square is (10a+5)2=100a(a+1)+25; the last two digits are always 25 and the preceding part equals a(a+1).
Pythagorean triplets from (2m, m^2-1, m^2+1): For any integer m > 1, (2m)2+(m2−1)2=(m2+1)2 gives the triple; examples include (6,8,10) for m=3 and (8,15,17) for m=4. The well-known triple (5,12,13) is not generated by this exact form but is another example.
Prime factorization approach: If N=∏p<em>ie</em>i, then N=∏p<em>ie</em>i/2 when all exponents e_i are even. To convert a non-square into a square, you can multiply by missing factors to pair up primes (or divide by the unpaired factor to reduce to a square, when possible).
Decimal square root method: The bar-method can be extended to decimals by placing bars on the decimal part as well; the quotient includes decimal digits correspondingly (e.g., 12.25=3.5).