Laws of Signs in Mathematics

The ley de signos suma y resta and Regla de los signos multiplicación y división are fundamental principles in mathematics that govern operations with positive and negative numbers. This comprehensive guide outlines the rules for addition, subtraction, multiplication, division, exponents, and roots of signed numbers.

Addition and Subtraction

The Reglas de los signos suma y resta dictate how to handle positive and negative numbers in addition and subtraction:

• When adding two positive numbers, the result is positive.
• When adding two negative numbers, the result is negative.
• When adding a positive and a negative number, or subtracting any number, the sign of the result follows the Sign of the Greater Absolute Value (SVM).

Example: (+2) + (+5) = +7, (-3) + (-8) = -11, (+9) + (-5) = +4, (-12) + (+8) = -4

Highlight: In addition and subtraction, the sign of the greater absolute value (SVM) determines the final sign of the result.

For subtraction:

• Subtracting a positive number is equivalent to adding its negative.
• Subtracting a negative number is equivalent to adding its positive.

Example: (+10) - (+4) = +6, (-4) - (-12) = +8, (-12) - (+3) = -15, (+2) - (-8) = +10

Multiplication and Division

The Ley de signos multiplicación and division rules are straightforward:

• Multiplying or dividing two numbers with the same sign results in a positive number.
• Multiplying or dividing two numbers with different signs results in a negative number.

Example: Multiplication: (+4)(+5) = +20, (-5)(-3) = +15, (+3)(-8) = -24, (-8)(+7) = -56

Example: Division: (+28) ÷ (+2) = +14, (-12) ÷ (-4) = +3, (-65) ÷ (+5) = -13, (+28) ÷ (-7) = -4

Exponents and Roots

For exponents:

• Any number (positive or negative) raised to an even power results in a positive number.
• A positive number raised to any power remains positive.
• A negative number raised to an odd power remains negative.

Example: (+4)² = +16, (-5)² = +25, (+2)³ = +8, (-3)⁴ = +81

For roots:

• Even roots of positive numbers have two solutions: positive and negative.
• Even roots of negative numbers result in imaginary numbers.
• Odd roots of positive numbers are positive.
• Odd roots of negative numbers are negative.

Vocabulary: Imaginary number - A complex number that can be written as a real number multiplied by the imaginary unit i, which is defined as the square root of -1.

These rules form the foundation of algebraic operations and are crucial for solving more complex mathematical problems involving signed numbers.