Problem 70 - Thinking Mathematically

Question

Express each expanded form as a Hindu-Arabic numeral. \(\left(7 \times 10^{4}\right)+\left(5 \times 10^{-3}\right)\)

Step-by-Step Solution

Answer

The Hindu-Arabic numeral form of \(\left(7 \times 10^{4}\right)+\left(5 \times 10^{-3}\right)\) is 70000.005

Step 1: Converting the first term

The first term is \(7 \times 10^{4}\). It can be converted into Hindu-Arabic numeral by calculating it as \(7 * 10000 = 70000\). Thus, the Hindu-Arabic numeral for the first term is 70000.

Step 2: Converting the second term

The second term is \(5 \times 10^{-3}\). It can be converted into Hindu-Arabic numeral by calculating it as \(5 * 0.001 = 0.005\). Thus, the Hindu-Arabic numeral for the second term is 0.005.

Step 3: Adding the converted terms

Add the Hindu-Arabic numerals obtained from the first and second terms to get the final result. That is: \(70000 + 0.005 = 70000.005\)

Key Concepts

Place Value

Place value is a fundamental concept in understanding numbers, especially when working with Hindu-Arabic numerals. In the decimal system, each digit of a number has a value based on its position. This position is a power of 10, making decimal numbers easy to multiply and divide by 10.
For example, in the number 7,0000.005:

• The digit 7 is in the ten-thousands place, which corresponds to \(10^4\), or 10,000. Therefore, its place value is \(7 \times 10^4 = 70,000\).
• The digit 5, located after the decimal point, is in the thousandths place, which corresponds to \(10^{-3}\), or 0.001. Consequently, its place value is \(5 \times 10^{-3} = 0.005\).

Understanding place value allows us to build and break down numbers, making every digit in our decimal system significant, whether it is before or after the decimal point.

Expanded Form

An expanded form breaks down a number to showcase the value of each digit according to its place value. This approach can help in understanding how numbers are constructed and how each digit contributes to the overall value.
Let’s take the expression from the exercise:

• The number expressed as \((7 \times 10^{4}) + (5 \times 10^{-3})\) is an expanded form where each part represents the value of the digit based on its place.
• The first term, \(7 \times 10^4\), highlights the value of 70,000, while \(5 \times 10^{-3}\) represents the 0.005 value.

By expressing numbers in expanded form, it becomes clearer how each segment of the number (both whole parts and decimals) contributes distinctively to the number as a whole. This breakdown is especially useful in mathematical operations where precision matters.

Decimal Notation

Decimal notation is our standard way of writing numbers that includes whole numbers and fractions within the same number. It stems from the flexibility of the base-ten system, which allows seamless transition and calculation from whole numbers into fractions, expressed as decimal points.
For instance, consider 70000.005:

• This number combines both an extensive whole number segment (70,000) and a fractional decimal part (0.005) effortlessly.
• The decimal point serves as a partition, indicating that numbers to the left are whole numbers and those to the right are fractions expressed in tenths, hundredths, thousandths, etc.

This notation not only makes numbers easy to read but also provides a convenient way to perform arithmetic operations, making it essential for everyday calculations.