Powers of ten are the building blocks of our base-ten number system. Every place in a number is worth 10 times as much as the place to its right and one tenth as much as the place to its left. This pattern lets us write very large and very small quantities using the same digits.
Place value matters because it controls the actual value of each digit, not just the digit itself.
The ones place is 10^0, which equals 1, and all other places are organized around it. Moving left multiplies by 10, giving tens, hundreds, thousands, and beyond. Moving right divides by 10, giving tenths, hundredths, thousandths, and smaller decimal places.
Understanding this structure makes it easier to compare numbers, shift decimals, use scientific notation, and read measurements accurately.
Key Facts
- 10^0 = 1, so the ones place is the center of the base-ten place value system.
- Moving one place left multiplies a value by 10.
- Moving one place right divides a value by 10.
- The value of a digit equals digit x place value.
- 10^3 = 1000 and 10^-3 = 0.001.
- In scientific notation, a number is written as a x 10^n, where 1 <= a < 10.
Vocabulary
- Place value
- Place value is the value of a digit based on its position in a number.
- Power of ten
- A power of ten is a number written as 10 raised to an exponent, such as 10^2 or 10^-4.
- Exponent
- An exponent tells how many times a base is multiplied by itself, or how many place shifts are made for powers of ten.
- Decimal point
- The decimal point separates whole-number places on the left from fractional places on the right.
- Scientific notation
- Scientific notation is a compact way to write very large or very small numbers using a decimal number multiplied by a power of ten.
Common Mistakes to Avoid
- Treating 10^0 as 0 is wrong because any nonzero number raised to the zero power equals 1.
- Counting decimal moves in the wrong direction is wrong because multiplying by 10 shifts digits left in place value, while dividing by 10 shifts digits right.
- Ignoring zeros as placeholders is wrong because zeros can hold places that determine the size of a number, such as the zero in 4,005.
- Reading 0.04 as four tenths is wrong because the 4 is in the hundredths place, so 0.04 means four hundredths.
Practice Questions
- 1 Write the value of the digit 7 in 58,742 and express that value as a power of ten times a digit.
- 2 Convert 0.0063 into scientific notation and identify the power of ten used.
- 3 Explain why moving a digit two places to the left in a place value chart makes its value 100 times larger.