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Chapter 1: Problem 14

For any integer, rational, or real number a the following equation holds: \(-a=(-1) \cdot a\). This proposition may seem way too obvious to you, but it's actually saying something very significant. \("-a "\) denotes the additive inverse of \(a\), while " (-1)\(\cdot a "\) denotes the additive inverse of 1 times the number \(a\). There is no a priori reason why theses two things should be the same. Try to think back to when you first learned this arithmetic stuff-at that time, it probably wasn't as obvious as it seems now. The exercise shows that it actually follows from even more basic facts about arithmetic.

Short Answer

Expert verified
Both \(-a\) and \(-1 \cdot a\) denote the same additive inverse of \(a\).

Step by step solution

01

Define the Additive Inverse

The additive inverse of a number is a value which, when added to the original number, yields zero. For a number \(a\), its additive inverse is denoted as \(-a\) such that \(a + (-a) = 0\).
02

Define Multiplication by -1

Multiplying any number by \(-1\) is equivalent to finding its additive inverse using a different method. If \(a\) is multiplied by \(-1\), it produces the natural definition of an additive inverse, so \(-1 \cdot a\) should equal the additive inverse of \(a\).
03

Use Distributive Property

Consider the distributive property of multiplication over addition: \(-1 \cdot (a + b) = (-1 \cdot a) + (-1 \cdot b)\). Let \(b = 0\). Thus, \(-1 \cdot (a + 0) = (-1 \cdot a) + (-1 \cdot 0)\).
04

Simplify Using the Identity Property

Using the identity property of zero in addition, \(a + 0 = a\) and thus \(-1 \cdot a = -a\). This shows that the results of both methods for finding the additive inverse are indeed equivalent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

additive inverse
The concept of the additive inverse is all about finding a number that when added to the original number results in zero. Think of it as the 'opposite' number. For any number \( a \), its additive inverse is denoted as \( -a \). This means that when you add them: \( a + (-a) = 0 \).
This property is very useful in simplifying expressions and solving equations. Knowing this helps us understand that every number has a counterpart that, when combined, results in zero.
distributive property
The distributive property is an important rule in algebra that connects multiplication and addition. This property states that for any numbers \( a, b, \textrm{ and } c \), the following holds: \[ a \times (b + c) = (a \times b) + (a \times c) \]
It helps us break down complex multiplication into simpler parts.
In the context of our exercise, we applied the distributive property to multiplication by \( -1 \) to show equivalence: \[ -1 \times (a + 0) = (-1 \times a) + (-1 \times 0) \]
identity property of zero
The identity property of zero is a fundamental rule that states adding zero to any number leaves the number unchanged.
Mathematically, this means \[ a + 0 = a \]
This property might seem very simple, but it is crucial in understanding other properties and performing arithmetic operations.
In our exercise, we used this property to simplify our expression: \[ -1 \times (a + 0) = -1 \times a \]
multiplication by -1
Multiplying any number by \( -1 \) is a straightforward way to find its additive inverse.
When you multiply \( a \) by \( -1 \), the result is the opposite of \( a \), or \( -a \): \[ -1 \times a = -a \]
This mirrors the definition of the additive inverse and shows that these two methods of finding the inverse are equivalent. Understanding this gives us another way to see that \( -a \) is simply \( -1 \) times \( a \).

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Most popular questions from this chapter

Given the expression: \((((a-b)+b)+b)(a-b)+b^{2}\) (a) Simplify the expression using the associative law ONLY. (b) Simplify the expression using the associative and distributive laws ONLY. (c) Simplify the expression using the associative, distributive, and commutative laws.
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