Problem 24 - Prealgebra

Question

Find each of the following products. $$\begin{array}{r} 12.345 \\ \times 1.000 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The product is 12.345.

Step 1: Understand the Problem

We want to multiply a decimal number, 12.345, by a whole number, 1.000.

Step 2: Remember the Identity Property of Multiplication

The identity property of multiplication states that any number multiplied by 1 remains unchanged. Thus, when you multiply any number by 1.000, the result should also be 12.345.

Step 3: Perform the Multiplication

Multiply 12.345 by 1.000. Simply align the decimal digits and perform the multiplication normally. Note that 1.000 is the same as 1.

Step 4: Write the Product

The correct product is obtained by maintaining the value of the original number because 1.000 acts as a multiplicative identity. Therefore, the product of 12.345 and 1.000 is 12.345.

Key Concepts

Decimal Numbers

Decimal numbers are numbers that are expressed using a decimal point to separate the whole number part from the fractional part. They are incredibly useful in mathematics to express values that are not whole numbers. For instance, in the problem, we have the decimal number 12.345. Here, 12 is the whole number part, and 345 is the fractional part.

Decimal numbers can be represented in various forms, such as:

• Standard form, like 2.5 or 7.89
• Expanded form, where a number like 34.56 is broken down into (3 x 10) + (4 x 1) + (0.5 x 10^-1) + (0.06 x 10^-2)

Understanding decimals is important because they allow us to represent fractions and perform precise calculations. When it comes to operations with decimal numbers, especially multiplication, aligning the decimal points can ensure accurate results. Even when multiplying by numbers like 1.000, which might seem like a whole number, it's crucial to line up the decimal points properly to maintain accuracy.

Identity Property

The identity property of multiplication is a fundamental concept in mathematics. This property states that any number multiplied by 1 results in the original number. For example, anything multiplied by 1, such as 7 or 12.345, remains unchanged. This property is extremely useful for simplification and verification of calculations.

Here's how the identity property can be used:

• Checking work, especially in complex or lengthy calculations, as it reassures that values remain consistent and correct.
• In algebra, it helps in solving equations, simplifying expressions, and understanding how values can remain constant.
• It plays an essential role in mathematical proofs, reinforcing the base logic of equality and operations.

When using the identity property, it's crucial to recognize that the number 1 can appear as 1.000, 1.00, or even just 1, in the context of decimals. The underlying principle remains the same, ensuring the integrity of any mathematical operation.

Multiplicative Identity

The concept of multiplicative identity closely ties into the identity property. It explicitly refers to the number 1, which is considered the 'multiplicative identity' because any number multiplied by 1 results in the original number. This is why, in the given problem, multiplying 12.345 by 1.000 gives us back 12.345.

Understanding multiplicative identity is crucial for:

• Recognizing how it simplifies multiplication tasks, ensuring no change to the original number.
• Ensuring accuracy in calculations involving decimals, especially when handling precise measurements or scientific data.
• Forming a foundational understanding of more complex mathematical concepts, such as identity elements in different number systems.

It's key to appreciate that regardless of how the number 1 is written, whether as 1.000 or just 1, the effect of multiplication remains the same. This property is vital in maintaining the stability and consistency of mathematical operations, particularly in diverse contexts involving decimals and whole numbers alike.