NYSTCE Mathematics CST (004)

Signed numbers include positive numbers, negative numbers, and zero. Understanding operations with signed numbers is foundational for algebraic manipulation.

The product of two signed numbers follows these rules:

The reasoning behind these rules stems from the properties of the real number system. When multiplying two negative numbers, think of it as reversing a reversal, which returns to the original direction (positive).

Division follows identical sign rules:

When multiplying or dividing signed numbers:

Same signs → Positive result

Different signs → Negative result

When multiplying more than two signed numbers, count the negative factors:

Even number of negative factors → Positive product

Odd number of negative factors → Negative product

Any number multiplied by zero equals zero: a × 0 = 0

Zero divided by any nonzero number equals zero: 0 ÷ a = 0 (where a ≠ 0)

Division by zero is undefined: a ÷ 0 is undefined

The absolute value |a| represents the distance from zero on the number line, always yielding a non-negative result.

When computing products or quotients:

Determine the sign using the rules above

Multiply or divide the absolute values to find the magnitude

Apply the determined sign to the result

Natural numbers (counting numbers) are the set {1, 2, 3, 4, 5, ...}. Some definitions include zero, written as {0, 1, 2, 3, ...}, called whole numbers.

Closure Property Natural numbers are closed under addition and multiplication:

If a and b are natural numbers, then a + b is a natural number

If a and b are natural numbers, then a × b is a natural number

Natural numbers are NOT closed under subtraction or division (results may not be natural numbers).

Addition: a + b = b + a

Multiplication: a × b = b × a

Addition: (a + b) + c = a + (b + c)

Multiplication: (a × b) × c = a × (b × c)

Distributive Property Multiplication distributes over addition and subtraction:

a × (b + c) = (a × b) + (a × c)

a × (b − c) = (a × b) − (a × c)

Additive identity: a + 0 = a

Multiplicative identity: a × 1 = a

A natural number a is divisible by natural number b if a ÷ b yields a natural number (no remainder).

Divisible by 2: Last digit is even (0, 2, 4, 6, 8) Divisible by 3: Sum of digits is divisible by 3 Divisible by 4: Last two digits form a number divisible by 4 Divisible by 5: Last digit is 0 or 5 Divisible by 6: Divisible by both 2 and 3 Divisible by 8: Last three digits form a number divisible by 8 Divisible by 9: Sum of digits is divisible by 9 Divisible by 10: Last digit is 0

The GCF of two or more numbers is the largest number that divides all of them evenly.

Method: Prime factorization approach

Find prime factorization of each number

Identify common prime factors with lowest powers

Multiply these together

The LCM of two or more numbers is the smallest number divisible by all of them.

Method: Prime factorization approach

Find prime factorization of each number

Take each prime factor with its highest power

Multiply these together

Relationship: For any two natural numbers a and b: GCF(a, b) × LCM(a, b) = a × b