Problem 28 - Thinking Mathematically

Question

Express each expanded form as a Hindu-Arabic numeral. \(\begin{aligned}\left(7 \times 10^{6}\right)+(&\left.4 \times 10^{5}\right)+\left(2 \times 10^{4}\right)+\left(3 \times 10^{3}\right) \\\\+&\left(1 \times 10^{2}\right)+\left(9 \times 10^{1}\right)+(6 \times 1) \end{aligned}\)

Step-by-Step Solution

Answer

The Hindu-Arabic numeral form of the expression is 7421096.

Step 1: Calculate each term individually

Each term of the expression needs to be evaluated one by one. For instance, 7*10^6 equals 7000000, 4*10^5 equals 400000, 2*10^4 equals 20000, 3*10^3 equals 3000, 1*10^2 equals 100, 9*10^1 equals 90, and finally 6*1 equals 6.

Step 2: Sum the results

Now, add the results from step 1 together. 7000000 + 400000 + 20000 + 3000 + 100 + 90 + 6 equals 7421096.

Key Concepts

Expanded Form Notation

When we write numbers, the expanded form notation is a way of breaking down and expressing the number to show the value of each digit according to its position, or place value. For example, the number 7421096 can be written in expanded form as:

\[\begin{equation}(7 \times 10^6) + (4 \times 10^5) + (2 \times 10^4) + (3 \times 10^3) + (1 \times 10^2) + (9 \times 10^1) + (6 \times 1)\end{equation}\]
Each term in the parentheses represents a specific place value, with the exponent of 10 signifying the position of the digit. The digit itself is multiplied by 10 to the power of its place value, laying out clearly how each digit contributes to the overall number. This form is particularly helpful for understanding the magnitude of numbers and for performing arithmetic operations, such as addition or multiplication.

Exponential Notation

Exponential notation is a mathematical shorthand that makes it easier to write and work with large numbers, such as those found in our example. In exponential notation, we use a base number (usually 10 in the Hindu-Arabic numeral system) and raise it to a certain power or exponent. For instance,10 to the power of 6 is written as \(10^6\).This essentially means multiplying 10 by itself 6 times.

In our expanded form example, each term involves a number in exponential form, such as \(7 \times 10^6\) for 7 million. By using exponential notation, we can comfortably write and comprehend very large (or small) numbers without having to write out a significant number of zeros, avoiding possible errors and simplifying calculations.

Place Value

The concept of place value is foundational in understanding the Hindu-Arabic numeral system. It refers to the value a digit has by virtue of its position within a number. Each position in a number has a different value, and the same digit can represent different amounts depending on where it's located.

In our example number, 7421096, the first digit '7' has a place value of 7 million because it is in the million's place. Similarly, the second digit '4' has a place value of four hundred thousand or \(4 \times 10^5\), and so on for each successive digit. Understanding place value is crucial for performing arithmetic operations and for grasping the relationship between numbers in different positions within a larger number.