Question

Identify the rule(s) of algebra illustrated by the statement.\((z-2)+0=z-2\)

Short answer

The rules of algebra illustrated by the statement are the 'Addition of Zero' rule and 'Identity' rule.

Step-by-step solution

Identify the algebraic rule

Looking at the expression, we see that the number \( z - 2 \) is being added to zero, and the result is the same number \( z - 2 \). This directly corresponds to the Addition of zero rule and the identity rule in Algebra. Thus, the rule illustrated by the statement is both 'Addition of Zero' and 'Identity'.

Key concepts

Addition of Zero

Understanding the addition of zero within the realm of algebra can greatly simplify the process of solving equations. This concept is commonly referred to as the zero property of addition. According to this property, when you add zero to any number or algebraic expression, the sum remains unchanged. It is akin to adding nothing at all, and the original value is preserved.

For example, consider the simple arithmetic expression \( 5 + 0 = 5 \) Here, adding zero did not alter the value of 5. The same rule applies to algebraic expressions, like \( (z - 2) + 0 = z - 2 \) as seen in the given exercise. No matter how complex the expression, zero maintains its neutral role, allowing the original expression to stand as is.

In addition to simplifying calculations, this principle is also a foundation for further algebraic concepts and operations, making it essential for students to grasp early on.

Identity Rule

The identity rule in algebra is closely related to the addition of zero, but it also extends to multiplication. The rule states that any number added to zero remains the same, which is essentially the definition of an additive identity. Similarly, any number multiplied by one keeps its identity, meaning one is the multiplicative identity.

These identity elements are crucial in algebra, as they do not change the value of the numbers they're combined with. For instance, if you see an expression like \( x + 0 \) or \( y \times 1 \) you instinctively know that the identity of \( x \) or \( y \) remains intact. Through this rule's lens, the exercise statement \( (z - 2) + 0 = z - 2 \) exemplifies the additive identity property: adding zero keeps the identity of \( z - 2 \). Understanding these identities helps students maintain the integrity of equations during transformations and simplifications.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operations but doesn't form a complete sentence – meaning it doesn't have an equals sign like an equation does. These expressions can range from simple, such as \( 2x + 3 \), to complex ones involving exponentiation or radicals.

The beauty of algebraic expressions lies in their ability to generalize problems and represent countless possible numbers through variables. When working with algebraic expressions, recognizing the rules for addition and multiplication, such as the identity rule, becomes incredibly beneficial. The expression given in our exercise, \( z - 2 \), is a modest illustration: it shows a variable \( z \) decreased by 2. However, as we have seen through applying the properties of addition of zero, even when combined with a neutral element like zero, their form persists without altering their value.