Chapter Notes: Number, Square Numbers

Square Numbers

• Square numbers are formed by multiplying a number by itself.
• Example: 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9, 4 × 4 = 16.
• Notation: 4 × 4 = 4² = 16.

Square numbers can be represented visually as squares made of smaller squares:

• Example: 1 = 1 × 1 (1 square), 4 = 2 × 2 (4 squares in a 2×2 grid), 9 = 3 × 3 (9 squares in a 3×3 grid).

Adding consecutive odd numbers produces square numbers:

• Example: 1 = 1².
• Example: 1 + 3 = 4 = 2².
• Example: 1 + 3 + 5 = 9 = 3².
• Example: 1 + 3 + 5 + 7 = 16 = 4².

Factor pairs identify square numbers:

• A number is a square number if it has a factor pair where both factors are equal.
• Example: 16 has factor pairs (1, 16), (2, 8), (4, 4); since 4 × 4 = 16, 16 is a square number.
• Example: 9 has factor pairs (1, 9), (3, 3); since 3 × 3 = 9, 9 is a square number.
• Sequence of square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ....

Triangular Numbers

• Triangular numbers form a pattern where each number can be represented as a triangular arrangement of dots or objects.
• Example: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ....

Triangular numbers are formed by adding consecutive natural numbers:

• 1 = 1.
• 1 + 2 = 3.
• 1 + 2 + 3 = 6.
• 1 + 2 + 3 + 4 = 10.
• 1 + 2 + 3 + 4 + 5 = 15.

Some numbers can be both square and triangular:

• Example: 1 = 1² (square) and 1 (triangular).
• Example: 36 = 6² (square) and 1 + 2 + 3 + 4 + 5 + 6 = 36 (triangular).

Adding pairs of triangular numbers can produce square numbers:

• Example: 1 + 3 = 4 = 2².

Tests of Divisibility

• Tests of divisibility determine whether a number is divisible by another without performing division.

Divisibility by 2:

• A number is divisible by 2 if the ones digit is even (0, 2, 4, 6, 8).
• Example: 1,736 is divisible by 2 because the ones digit is 6.

Divisibility by 4:

• A number is divisible by 4 if the last two digits form a number divisible by 4.
• Example: 1,736 is divisible by 4 because 36 ÷ 4 = 9.
• Since 100 = 4 × 25, every hundred is divisible by 4, so only the last two digits need checking.

Divisibility by 8:

• A number is divisible by 8 if the last three digits form a number divisible by 8.
• Example: 1,736 is divisible by 8 because 736 ÷ 8 = 92.
• Since 1,000 = 8 × 125, every thousand is divisible by 8, so only the last three digits need checking.

Prime Numbers

• Prime numbers have exactly two divisors: 1 and themselves.
• Example: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...
• 1 is not a prime number because it has only one divisor (1).
• Composite numbers have more than two divisors.
• Example: 4 (divisors: 1, 2, 4), 6 (divisors: 1, 2, 3, 6).

The Sieve of Eratosthenes identifies prime numbers up to 100:

• Start with a grid from 1 to 100.
• Cross out 1 (not prime).
• Circle 2, cross out all other multiples of 2 (4, 6, 8, ...).
• Circle 3, cross out all other multiples of 3 (6, 9, 12, ...).
• Skip 4 (already crossed out), circle 5, cross out multiples of 5, and continue.
• Remaining numbers are prime: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Arrays show prime numbers have only two divisors:

• Example: 3 can be arranged as 3 × 1 or 1 × 3, confirming divisors 1 and 3.
• Not all prime numbers are odd (e.g., 2 is prime), and not all odd numbers are prime (e.g., 9 is composite).