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Chapter 4: Problem 21

In Exercises 9-22, write each Hindu-Arabic numeral in expanded form. \(230,007,004\)

Short Answer

Expert verified
The expanded form of the number \(230,007,004\) is \[200,000,000 + 30,000,000 + 7,000 + 4\].

Step by step solution

01

Identify the Place Values

The number \(230,007,004\) has eight digits, so it falls into the 'hundreds of millions' category. Starting from the right to left, the places are: units, tens, hundreds, thousands, ten thousands, hundred thousands, millions, tens of millions, and finally hundreds of millions. Therefore, in the given number, the digit 2 is in the hundreds of millions place, 3 is in the ten millions place, 0 is in the millions place, 0 is in the hundred thousands place, 0 is in the ten thousands place, 7 is in the thousands place, 0 is in the hundreds place, 0 is in the tens place, and finally the digit 4 is in the units place.
02

Write Each Digit in Expanded Form

Now write down each digit in the number along with its corresponding place value. In expanded form, this becomes \[200,000,000 + 30,000,000 + 7,000 + 4\].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expanded Form
The concept of expanded form is a crucial aspect of understanding numbers in elementary mathematics. When you express a number in its expanded form, you break it down into the sum of its individual digits, each multiplied by its place value.
This process helps in visualizing the role of each digit within a larger number. For example, the number 230,007,004 is expressed in expanded form as \(200,000,000 + 30,000,000 + 7,000 + 4\). This representation clearly shows how each part contributes to the whole number.
Expanded form supports arithmetic operations and lays the foundation for more advanced mathematical concepts.
Place Value
Place value is the value assigned to the position of a digit within a number. It's the key to understanding numerical representation in the Hindu-Arabic numeral system. Each position in a number has a value ten times that of the position to its right.
Here is a breakdown of the power of each position:
  • Units: \(10^0\)
  • Tens: \(10^1\)
  • Hundreds: \(10^2\)

This pattern continues with thousands, ten thousands, and so forth.
In our example, 230,007,004, the digit 2 holds the place value of hundreds of millions, highlighting how place value helps in understanding a number's magnitude.
Numerical Representation
Numerical representation involves expressing numbers in different forms to convey information clearly and effectively. The Hindu-Arabic numeral system is widespread, utilizing 10 symbols (0-9) and place value to denote any quantity.
This system is efficient for various mathematical operations, providing a clear structure for both small and very large numbers.
When representing a number like 230,007,004, we use a sequence of digits, leveraging place value to give each digit meaning.
Elementary Mathematics
Elementary mathematics forms the foundation for all higher-level math. It begins with basic arithmetic鈥攁ddition, subtraction, multiplication, and division鈥攁nd extends to concepts like place value and fractions.
Understanding place value and numerical representation is essential at this level. These concepts enable students to grasp more complex ideas such as decimals and fractions later on.
  • Place value clarifies the importance of digit positioning.
  • Expanded form decomposes numbers to simplifiy operations.

These building blocks enhance problem-solving abilities and logical thinking, vital skills for mathematical literacy.

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Most popular questions from this chapter

In Exercises 25鈥�34, multiply in the indicated base. $$ \begin{array}{r} 543_{\text {seven }} \\ \times \quad 5_{\text {seven }} \\ \hline \end{array} $$

In Exercises 19-32, mentally convert each base ten numeral to a numeral in the given base. 12 to base seven

In Exercises 65-66, break each binary sequence into groups of seven digits and write the word represented by the sequence. 101000010000011001100

Make Sense? In Exercises 57-60, determine whether each statement makes sense or does not make sense, and explain your reasoning. Arithmetic in bases other than ten works just like arithmetic in base ten.

In Exercises 43-48, write each Hindu-Arabic numeral as a traditional Chinese numeral. 2965
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