Chapter 1: Variables, Real Numbers, and Mathematical Models

Addition of Real Numbers

This section introduces the foundational concepts of adding real numbers, including the use of the number line, properties of addition, and applications in real-world contexts. Mastery of these concepts is essential for success in algebra and higher mathematics.

Using the Number Line to Find a Sum

• Number Line Method: To add two real numbers a and b using a number line:

  • Start at a on the number line.

  • If b is positive, move b units to the right.

  • If b is negative, move |b| units to the left.

  • If b is zero, remain at a.

  • The final position represents the sum a + b.

• Example: Find the sum 7 + (–9) using a number line:

  • Start at 7, move 9 units left, finish at –2.

• Example: Find the sum –1 + (–3):

  • Start at –1, move 3 units left, finish at –4.

• Example: Find the sum –5 + 3:

  • Start at –5, move 3 units right, finish at –2.

Properties of Addition

• Identity Property: For any real number a, a + 0 = a.

• Inverse Property: For any real number a, a + (–a) = 0.

Adding Two Numbers with the Same Sign

• Add the absolute values of the numbers.

• Use the common sign as the sign of the sum.

• Example: –10 + (–25) = –35

Adding Two Numbers with Different Signs

• Subtract the smaller absolute value from the greater absolute value.

• Use the sign of the number with the greater absolute value as the sign of the sum.

• Example: –15 + 2 = –13

• Example: –0.4 + 1.6 = 1.2

Simplifying Algebraic Expressions Using Addition Rules

• Combine like terms by adding their coefficients.

• Example: –20x + 3x = (–20 + 3)x = –17x

• Example: 3y + (–10z) + (–10y) + 16z = –7y + 6z

• Example: 5(2x + 3) + (–30x) = –20x + 15

Applications of Positive and Negative Numbers

• Positive and negative numbers are used to represent real-world quantities such as gains and losses, temperature changes, bank transactions, and elevation changes.

• Applied problems often involve a series of additions with both positive and negative values.

Series of Additions

• To add a series of positive and negative numbers:

  1. Add all positive numbers together.

  2. Add all negative numbers together.

  3. Add the two sums obtained in steps 1 and 2.

Example: Application of Adding Signed Numbers

• Problem: The water level of a reservoir changes over five months: +2 ft, –4 ft, +1 ft, –5 ft, +3 ft. What is the net change?

• Solution:

  • Sum: 2 + (–4) + 1 + (–5) + 3

  • Group positives: (2 + 1 + 3) = 6

  • Group negatives: (–4) + (–5) = –9

  • Final sum: 6 + (–9) = –3

  • Interpretation: The water level decreased by 3 feet over five months.