Problem 9 - Algebra 1: Concepts and Skills

Question

Copy and complete the proof of the statement: For all real numbers a and b, \((a+b)-b=a\) $$ \begin{aligned} (a+b)-b &=(a+b)+(-b) \\ &=a+[b+(-b)] \\ &=a+0 \\ &=a \end{aligned} $$ Definition of subtraction. Associative property of addition ? _____ ? _____

Step-by-Step Solution

Answer

The missing steps are: 1) (a+b)-b is the same as (a+b)+(-b) based on the definition of subtraction. 2) Then, we use the associative property of addition to change a+[b+(-b)] to a+(0), since b+(-b) equals zero. 3) Finally, we simplify a+(0) to a.

Step 1 - Definition of subtraction

In this first step we should understand that subtraction is the addition of the opposite. Therefore, (a+b)-b is the same as (a+b)+(-b)

Step 2 - Associative property of addition

The associative property, which states that the way numbers are grouped in an addition or multiplication operation does not change their sum or product, is applied in this step. Therefore, a+[b+(-b)] becomes a+(0), as b+(-b) equals zero.

Step 3 - Simplify

In this step we simplify the expression a+(0) to just a, using the property that adding zero to any number gives the same number.

Key Concepts

Definition of Subtraction

Subtraction might seem like a simple operation, but it's essential to understand it more deeply. In math, subtraction is defined as the addition of the opposite. This means when you subtract a number, you are actually adding its negative equivalent. For example, subtracting 5 from 8 is the same as adding -5 to 8.

So, when you see something like

• \((a+b)-b\)

understand that it can be rewritten as

• \((a+b)+(-b)\)

This view is important for performing operations like simplifying or solving equations. Once you recognize subtraction as adding a negative, using properties like the associative property becomes more natural.

Real Numbers

Real numbers are the foundation of our number system. They include all the numbers you can think of on the number line, covering everything from integers to fractions, and from negative numbers to positive numbers.

In simpler terms:

• Whole numbers (e.g., 0, 1, 2)
• Integers (e.g., -3, 0, 4)
• Rational numbers (fractions like \(\frac{2}{3}\) or \(-\frac{3}{4}\))
• Irrational numbers (like \(\pi\) or \(\sqrt{2}\))

These numbers are everywhere, and math often involves manipulating real numbers using various operations and properties. They are crucial in the proof mentioned, as understanding and recognizing the behavior of real numbers allow us to apply properties like associativity correctly.

Properties of Addition

One powerful tool in math is understanding and using properties of operations like addition. When adding numbers, several properties make computations easier:

• Associative Property: This property states that the grouping of numbers doesn't affect their sum. For any numbers \(a\), \(b\), and \(c\): \[(a+b)+c = a+(b+c)\]
• Commutative Property: The order in which you add numbers doesn't matter. For \(a\) and \(b\):\[a+b = b+a\]
• Identity Property: Adding zero to any number results in the same number. For any number \(a\): \[a+0 = a\]

In our proof, the associative property helps rearrange terms smoothly, showing how \((a+b)+(-b)\) simplifies to \(a+0\). Using these properties is essential for solving problems and simplifying expressions.