What Is The Place Value Of The Underlined Digit

What is the Place Value of the Underlined Digit? A Comprehensive Guide

Understanding place value is fundamental to mastering arithmetic and forming a strong foundation in mathematics. This comprehensive guide delves into the concept of place value, explaining it in detail and providing numerous examples to solidify your understanding. We will explore various number systems, including whole numbers, decimals, and even delve into the fascinating world of place value in different number bases. By the end of this article, you'll be able to confidently identify the place value of any underlined digit in any given number.

What is Place Value?

Place value refers to the value of a digit based on its position within a number. In the decimal number system (base-10), each position represents a power of 10. Moving from right to left, the positions are ones, tens, hundreds, thousands, and so on. Each position represents a progressively larger power of ten.

Think of it like this: Imagine a well-organized warehouse where each shelf represents a place value. The rightmost shelf holds single items (ones), the next shelf holds bundles of ten items (tens), the next shelf holds bundles of one hundred items (hundreds), and so on. The value of an item depends entirely on which shelf it's placed on.

The Decimal Number System (Base-10)

The decimal system is the most common number system used worldwide. It uses ten digits (0-9) and is based on powers of 10. Let's illustrate place value with an example:

Example: Consider the number 3<u>4</u>5,217. The underlined digit is 4.

To determine its place value, we examine its position:

• 7: Ones place (10⁰ = 1)
• 1: Tens place (10¹ = 10)
• 2: Hundreds place (10² = 100)
• 5: Thousands place (10³ = 1000)
• 4: Ten Thousands place (10⁴ = 10,000)
• 3: Hundred Thousands place (10⁵ = 100,000)

Therefore, the place value of the underlined digit 4 is ten thousands. The digit 4 represents 4 x 10,000 = 40,000.

Place Value of Decimals

The concept of place value extends seamlessly to numbers containing decimals. To the right of the decimal point, the positions represent tenths, hundredths, thousandths, and so on – negative powers of 10.

Example: Consider the number 12.<u>3</u>45

• 5: Thousandths place (10^-3 = 0.001)
• 4: Hundredths place (10^-2 = 0.01)
• 3: Tenths place (10^-1 = 0.1)
• 2: Ones place (10⁰ = 1)
• 1: Tens place (10¹ = 10)

The underlined digit 3 is in the tenths place, meaning its place value is tenths and its value is 3 x 0.1 = 0.3.

Place Value in Different Number Bases

While the decimal system (base-10) is prevalent, other number systems exist, each with its own unique place value system. For example:

Binary (Base-2)

The binary system uses only two digits (0 and 1) and is crucial in computer science. Place values are powers of 2.

Example: The binary number 1<u>0</u>11₂ (the subscript 2 indicates base 2).

• 1: 2³ = 8
• 0: 2² = 4
• 1: 2¹ = 2
• 1: 2⁰ = 1

The underlined digit 0 is in the 2² place, so its place value is four. However, its value in this number is 0 x 4 = 0.

Octal (Base-8)

The octal system utilizes eight digits (0-7) and its place values are powers of 8.

Example: 3<u>7</u>2₈

• 2: 8⁰ = 1
• 7: 8¹ = 8
• 3: 8² = 64

The place value of the underlined digit 7 is eight. Its value in the number is 7 x 8 = 56.

Hexadecimal (Base-16)

The hexadecimal system employs sixteen digits (0-9 and A-F, where A=10, B=11, C=12, D=13, E=14, F=15). Place values are powers of 16.

Example: <u>B</u>F2₁₆

• 2: 16⁰ = 1
• F (15): 16¹ = 16
• B (11): 16² = 256

The place value of the underlined digit B is 256. Its value in the number is 11 x 256 = 2816.

Advanced Applications of Place Value

Understanding place value is not just about identifying the position of a digit; it forms the bedrock for various mathematical operations:

• Addition and Subtraction: Correctly aligning digits based on their place value is crucial for accurate addition and subtraction.
• Multiplication and Division: Understanding place value helps in comprehending the shifting of digits during multiplication and division.
• Rounding and Estimation: Place value plays a critical role in rounding numbers to a specific place value.
• Scientific Notation: Scientific notation utilizes place value to represent extremely large or small numbers concisely.

Troubleshooting Common Mistakes

Students often encounter difficulties when dealing with place value, particularly with decimals and different number bases. Here are some common pitfalls and how to avoid them:

• Confusion with Digit Value vs. Place Value: Remember that the digit value is the number itself (e.g., the digit 7), while the place value represents the position's magnitude (e.g., ones, tens, hundreds).
• Incorrectly Identifying Decimal Places: Pay close attention to the position of the decimal point when determining the place value of digits to its right.
• Misunderstanding Number Bases: When working with different number bases, ensure you understand the powers of the base being used.

Practice Exercises

Let's test your understanding with some practice exercises:

1. What is the place value of the underlined digit in 2<u>8</u>,345?
2. What is the place value of the underlined digit in 1.0<u>5</u>7?
3. What is the place value of the underlined digit in 1<u>1</u>01₂?
4. What is the place value of the underlined digit in <u>A</u>2F₁₆?
5. What is the place value of the underlined digit in 5<u>6</u>78₈?

Answers:

1. Thousands
2. Hundredths
3. Four
4. 4096
5. Eight

Conclusion

Understanding place value is a cornerstone of mathematical proficiency. It's a fundamental concept that underlies many arithmetic operations and extends into more advanced mathematical concepts. By mastering place value in the decimal system and different number bases, you'll build a solid mathematical foundation and unlock a deeper understanding of numbers and their representation. Practice is key to solidifying your knowledge, so continue working through examples and exercises to improve your comprehension and confidence. Remember to carefully analyze the position of the digit within the number to accurately determine its place value. This guide provides a solid starting point, and with continued practice, you'll become adept at identifying the place value of any underlined digit.