Class 7 Mathematics Ch 2 Mastery Guide: Square and Square Roots, Primes, Factorization & Real-World Applications (FBISE)

Section 1: Number Types, Primes, Sieve of Eratosthenes & Twin Primes

Mathematics builds upon the fundamental classification of natural numbers $\mathbb{N} = \{1, 2, 3, 4, 5, \dots\}$:

• Even Numbers: Natural numbers divisible by 2 without remainder (multiples of 2). A number is even if its units digit is $0, 2, 4, 6,$ or $8$. Examples: $2, 4, 6, 8, 10, 56798, 911756$.
• Odd Numbers: Natural numbers not divisible by 2. A number is odd if its units digit is $1, 3, 5, 7,$ or $9$. Examples: $1, 3, 5, 7, 9, 798125, 3219$.

Algebraic Properties of Even and Odd Numbers:

Prime & Composite Numbers:

A natural number greater than 1 that has only two distinct factors (1 and itself) is called a prime number. Examples: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31$.
A natural number that has more than two different factors is called a composite number. Examples: $4, 6, 8, 9, 10, 12, 14, 15, 16, 18$.

The Sieve of Eratosthenes (276 B.C. – 194 B.C.):

To find all primes up to 100: 1. Write numbers 1 to 100 in 10 rows. Cross out 1.
2. Circle 2 (prime), then cross out all multiples of 2 ($4, 6, 8, \dots$).
3. Circle 3 (prime), then cross out all multiples of 3 ($6, 9, 12, \dots$).
4. Circle 5 (prime), cross out all multiples of 5; circle 7, cross out all multiples of 7.
5. The remaining circled numbers are the 25 prime numbers between 1 and 100:

Twin Primes & Prime Triplets:

• Twin Primes: Prime numbers that differ by exactly 2. Examples: $(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73)$.
• Prime Triplet: Three consecutive prime numbers that differ by 2. The set $\mathbf{(3, 5, 7)}$ is the only known prime triplet in all of mathematics!
• Goldbach's Representation: Every even number greater than 4 can be written as the sum of two odd primes (e.g. $12 = 5 + 7$, $36 = 17 + 19$, $60 = 29 + 31$).

Section 2: Squares of Numbers, Proper Fractions & Decimals

When a number $x$ is multiplied by itself, the result $x \times x = x^2$ is called the square of $x$. Geometrically, $x^2$ is the area of a square whose side length is $x$.

• Perfect Square of Natural Numbers: An integer is a perfect square if its prime factors can be grouped into exact pairs of equal factors. For example, $144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = (2^2) \times (2^2) \times (3^2) = (2 \times 2 \times 3)^2 = 12^2$, so 144 is a perfect square. But $80 = 2^4 \times 5$ has an unpaired factor 5, so 80 is NOT a perfect square.
• Square of Even & Odd Numbers:
  • The square of an even number is always even: $2^2 = 4$, $6^2 = 36$, $14^2 = 196$, $20^2 = 400$.
  • The square of an odd number is always odd: $1^2 = 1$, $5^2 = 25$, $11^2 = 121$, $17^2 = 289$.
• Square of Proper Fractions: When squaring a proper fraction $\frac{a}{b}$ ($a < b$): $$\left(\frac{a}{b}\right)^2 = \frac{a^2}{b^2}$$ Notice that $\frac{a^2}{b^2} < \frac{a}{b}$! For example: $\left(\frac{3}{5}\right)^2 = \frac{9}{25} = 0.36$, whereas $\frac{3}{5} = 0.60$. Since $0.36 < 0.60$, the square of a proper fraction is always less than the fraction itself!
• Square of Decimals Less than 1: Similarly, for any decimal number $0 < d < 1$, $d^2 < d$. For example, $(0.2)^2 = 0.04 < 0.2$; $(0.5)^2 = 0.25 < 0.5$; $(0.07)^2 = 0.0049 < 0.07$.

Section 3: Square Roots & The Prime Factorization Method

The square root of a number $y$ is a number $x$ which, when multiplied by itself, produces $y$: $$x^2 = y \iff x = \sqrt{y} = y^{1/2}$$ The symbol $\sqrt{\cdot}$ is called the radical sign, and the number beneath it is called the radicand.

While $(-4) \times (-4) = 16$ and $4 \times 4 = 16$, every positive real number has two square roots ($+4$ and $-4$). In Grade 7, we study the principal (positive) square root: $\sqrt{16} = 4$.

The 4-Step Prime Factorization Algorithm for Square Roots:

1. Step 1: Find prime factors: Divide the number by prime numbers ($2, 3, 5, 7, 11, \dots$) until the quotient is 1.
2. Step 2: Group identical factors into pairs: Write the factors in exponential form $p_1^2 \times p_2^2 \times \dots$.
3. Step 3: Apply the radical sign: Take one factor from each pair outside the radical sign: $\sqrt{p^2} = p$.
4. Step 4: Multiply the extracted factors: The product gives the exact square root!

The Fundamental Product Property:

Section 4: Square Roots of Common and Mixed Fractions

To find the square root of a common fraction $\frac{a}{b}$, apply the Quotient Rule: $$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$$ Find the prime factorization of the numerator $a$ and denominator $b$ independently, group factors into pairs, and take square roots separately.

Section 5: Square Roots of Decimal Numbers

To find the square root of a decimal fraction by prime factorization:

1. Convert the decimal into a common fraction: Remove the decimal point and place 1 followed by as many zeros in the denominator as there are decimal digits (e.g. $1.69 = \frac{169}{100}$, $0.0256 = \frac{256}{10000}$).
2. Take square root of numerator and denominator: $\sqrt{\frac{169}{100}} = \frac{\sqrt{169}}{\sqrt{100}} = \frac{13}{10}$.
3. Convert back to decimal form: $\frac{13}{10} = \mathbf{1.3}$.

Section 6: Real-World Applications & Word Problem Formulas

Square roots are ubiquitous in construction, land measurement, and physical arrangements: