Positive and negative numbers form the foundation of arithmetic and are crucial in understanding various mathematical concepts.The understanding of positive and negative numbers allows us to perform basic arithmetic operations, such as addition, subtraction, multiplication, and division. They also help describe real-world situations, from calculating profits and losses in business to measuring below-zero temperatures. Mastery of these numbers is essential for progressing in mathematics, as they serve as the building blocks for more complex concepts like algebra, calculus, and beyond.

• Positive Numbers: Numbers greater than zero, usually denoted by a plus (+) sign.
• Negative Numbers: Numbers less than zero, denoted by a minus (-) sign.

Operations with Positive and Negative Numense

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Note 1 : The exponent of a positive real number is always positive. The exponent of a negative real number is positive if the exponent is even and negative if the exponent is odd.

Note 2 : When a smaller number is subtracted from a number, the result is positive because the original number remains larger. For example: 5−3 = 25 – 3 = 25−3 =2.

However, when a larger number is subtracted, the result is negative because the subtracted number is greater. For example: 3−5 =−23 – 5 = -23− 5 = −2. This depends on the relative sizes of the numbers.

Powers of Positive and Negative Numbers

1. Powers of a Positive Number:

• Any positive number raised to any power is always positive. For example:
  • 2² = 4
  • 2³ = 8
  • 2⁴ = 16
• In other words, any positive number raised to any positive or negative power results in a positive value.

2. Powers of a Negative Number:

• Even Powers: The power of a negative number raised to an even number is always positive because multiplying two negative numbers results in a positive value.
  • (-2)² = 4
  • (-3)² = 9
• Odd Powers: The power of a negative number raised to an odd number is always negative because multiplying a negative number by itself an odd number of times results in a negative value.
  • (-2)³ = -8
  • (-3)³ = -27

Inequalities

Inequalities are mathematical expressions that indicate a relationship where two values are not equal, typically represented by symbols such as > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). These relationships help us understand the relative sizes of numbers and play a crucial role in mathematical analysis, functions, equations, and many other areas.

Examples:

• Numbers greater than 5 → x > 5
• Ali’s age is less than 20 → a < 20
• People aged 18 and above can obtain a driving license → e ≥ 18

Properties of Inequalities

Addition and Subtraction

If the same number is added or subtracted from both sides of an inequality, the direction of the inequality remains unchanged.

Multiplication and Division by Positive Numbers

If both sides of an inequality are multiplied or divided by the same positive number, the direction of the inequality remains unchanged.

Multiplication and Division by Negative Numbers

If both sides of an inequality are multiplied or divided by the same negative number, the direction of the inequality changes.

Note:
The direction of an inequality only changes when multiplied or divided by negative numbers; otherwise, the direction does not change.