Powers of 10 and place value to thousandths Knowledge-Based Study

Study notes

1. Place Values to the Right of the Decimal Point

   Our number system is a base-10 system. Each place value is ten times the value of the place to its immediate right, and one-tenth of the value of the place to its immediate left.

   When moving to the right of the decimal point, we partition whole units into fractional parts:

   • Tenths (0.1 or $\frac{1}{10}$): The first position to the right of the decimal point. For example, in 0.4, the digit 4 represents 4 tenths.
   • Hundredths (0.01 or $\frac{1}{100}$): The second position to the right of the decimal point. For example, in 0.07, the digit 7 represents 7 hundredths.
   • Thousandths (0.001 or $\frac{1}{1000}$): The third position to the right of the decimal point. For example, in 0.006, the digit 6 represents 6 thousandths.

   Consider the number 3.475:

   • 3 is in the ones place: $3 \times 1 = 3$
   • 4 is in the tenths place: $4 \times 0.1 = 0.4$
   • 7 is in the hundredths place: $7 \times 0.01 = 0.07$
   • 5 is in the thousandths place: $5 \times 0.001 = 0.005$

   Multiplying and Dividing by Powers of 10

   Powers of 10 describe numbers formed by multiplying 10 by itself, such as $10^1 = 10$, $10^2 = 100$, and $10^3 = 1000$.

   Multiplying or dividing a decimal by a power of 10 changes the place value of each digit:

   • Multiplying by 10, 100, or 1000: Each digit shifts to a place value that is 10, 100, or 1000 times greater. This makes the decimal point appear to move to the right by as many places as there are zeros in the power of 10.
     Example: $0.358 \times 10 = 3.58$
     Example: $0.358 \times 100 = 35.8$
     Example: $0.358 \times 1000 = 358$
   • Dividing by 10, 100, or 1000: Each digit shifts to a place value that is smaller by a factor of 10, 100, or 1000. This makes the decimal point appear to move to the left.
     Example: $6.4 \div 10 = 0.64$
     Example: $6.4 \div 100 = 0.064$
     Example: $6.4 \div 1000 = 0.0064$

   Writing Decimals in Expanded Form

   A decimal number can be represented as the sum of the values of each of its digits. For instance, the number 5.208 can be written in two standard expanded forms:

   • Decimal Expanded Form: $5 + 0.2 + 0.008$
   • Fractional Expanded Form: $(5 \times 1) + \left(2 \times \frac{1}{10}\right) + \left(8 \times \frac{1}{1000}\right)$

   Notice that because the hundredths digit is 0, it contributes 0 to the sum and can be omitted in standard expanded expressions.

   Key Misconceptions to Avoid

   • Confusing Tenths and Tens: Remember that tenths represent fractional parts smaller than one whole ($\frac{1}{10}$), whereas tens represent whole groups of ten.
   • Ignoring Placeholding Zeros: The number 0.05 is not equal to 0.5. The zero in the tenths place of 0.05 preserves the position of the 5 in the hundredths place.
   • Longer Does Not Mean Larger: For decimals, the count of digits after the decimal point does not determine size. For example, $0.4 > 0.375$ because 4 tenths is greater than 3 tenths.