2.1 Place Value System & Number Representations

2.1 Place Value System & Number Representations

Understanding the Base-Ten System

The foundation of modern arithmetic is the base-ten positional numeral system (also known as the decimal system). In a positional place value system, the value of a digit depends not only on the digit itself (from $0$ to $9$) but also on its position or "place" within the number relative to the decimal point.

The fundamental principle governing base-ten numeration is that each place value position represents a power of ten. Moving one position to the left increases a digit's value by a factor of $10$, while moving one position to the right decreases its value by a factor of $10$ (or multiplies it by $\frac{1}{10}$).

$$1 \text{ unit} \times 10 = 1 \text{ ten} \quad \text{and} \quad 1 \text{ tenth} \times \frac{1}{10} = 1 \text{ hundredth}$$

The Place Value Chart: Millions to Thousandths

For the Praxis 5003 exam, elementary educators must demonstrate mastery of place values ranging from the millions place down to the thousandths place. The chart below organizes these positions into periods (groups of three digits separated by commas in whole numbers).

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Forms of Number Representation

Numbers can be expressed in multiple equivalent forms. Elementary teacher candidates must be able to convert fluidly among standard form, word form, and expanded form.

1. Standard Form

Standard form is the conventional way of writing numbers using digits $0$ through $9$, with commas separating periods of three digits to the left of the decimal point.

Example: $4,052,809.043$

2. Written Form (Word Form)

Word form translates digits into written words. When writing or reading numbers in English, strict formal rules apply:

• Use hyphens for compound numbers between 21 and 99 (e.g., twenty-five, fifty-two).
• Do NOT use the word "and" when naming whole numbers. The word "and" is reserved exclusively to indicate the decimal point.
• State the period name after each group of three digits (except the units period).
• Name the decimal portion as a whole number, followed by the place value of the final digit to the right.

Example: $4,052,809.043$ is written as:

"Four million, fifty-two thousand, eight hundred nine and forty-three thousandths."

3. Expanded Form

Expanded form decomposes a number to show the value of each individual digit as an explicit sum. On the Praxis 5003, expanded form appears in two main formats:

Additive Value Format:

$$4,052,809.043 = 4,000,000 + 50,000 + 2,000 + 800 + 9 + 0.04 + 0.003$$

Multiplicative / Exponential Format:

$$(4 \times 10^6) + (5 \times 10^4) + (2 \times 10^3) + (8 \times 10^2) + (9 \times 10^0) + (4 \times 10^{-2}) + (3 \times 10^{-3})$$

Notice that positions containing the digit $0$ are typically omitted from the expanded sum because $0 \times 10^n = 0$.

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Powers of Ten & Multiplicative Relationships

Understanding place value requires recognizing how multiplying or dividing by powers of $10$ shifts digits relative to the decimal point.

Comparing Digit Values within Numbers

Because each place value is 10 times the value of the place to its right:

• A digit in the hundreds place is $10 \times 10 = 100$ times greater than the same digit in the ones place.
• A digit in the thousands place is $1,000$ times greater than the same digit in the tenths place ($10^3 \div 10^{-1} = 10^{3 - (-1)} = 10^4 = 10,000$).

Worked Example 1:

Compare the value of the digit $5$ in the number $54,300$ to the digit $5$ in the number $5.43$.

Solution:

1. Identify place values:
   • In $54,300$, the digit $5$ is in the ten-thousands place ($50,000$).
   • In $5.43$, the digit $5$ is in the ones place ($5$).
2. Calculate the ratio: $$\frac{50,000}{5} = 10,000 = 10^4$$
3. Conclusion: The digit $5$ in $54,300$ is $10,000$ times greater (or $10^4$ times greater) than the digit $5$ in $5.43$.

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Comparing & Ordering Whole Numbers and Decimals

To compare two or more numbers accurately, follow a structured, step-by-step place value alignment procedure.

Step-by-Step Comparison Strategy

1. Line up decimal points vertically: Append trailing zeros to the right of decimal numbers so all numbers have the same number of decimal places.
2. Compare starting from the leftmost non-zero place value: Move from left to right, column by column.
3. Identify the first point of difference: The number with the larger digit in this highest differing place value is the larger number.

Example: Compare 0.4, 0.385, 0.405, and 0.09
Step 1: Pad with trailing zeros to thousandths place:
        0.400
        0.385
        0.405
        0.090

Step 2: Compare tenths place:
        0.090 (tenths: 0)  --> Smallest
        0.385 (tenths: 3)  --> Next smallest
        0.400 & 0.405 (tenths: 4) --> Tie for largest!

Step 3: Break tie for 0.400 vs 0.405 using hundredths & thousandths:
        0.400 (hundredths: 0, thousandths: 0)
        0.405 (hundredths: 0, thousandths: 5) --> 0.405 is larger than 0.400

Final Ordered List (Least to Greatest): 0.09 < 0.385 < 0.4 < 0.405