The Shifting Point: Decimals and Powers of Ten

When we multiply a decimal by a power of ten (like $10$, $100$, or $1,000$), we aren't just adding zeros. We are actually shifting the place value of every digit in the number. The easiest way to track this is by moving the decimal point to the right. Why the right? Because moving the decimal to the right makes the number larger! The number of 'jumps' the decimal makes is exactly equal to the number of zeros in the power of ten. For example, multiplying by $10$ (one zero) means $1$ jump. Multiplying by $100$ (two zeros) means $2$ jumps.

When you multiply by $100$ or $1,000$, the same rule applies, but the 'slider' moves further. Multiplying by $100$ moves the decimal two places to the right. Multiplying by $1,000$ moves it three places. Sometimes, you might run out of digits! If you need to move the decimal but there are no more numbers, you must use a placeholder zero. This ensures the digits stay in their correct new place value columns.

It looks like the decimal is moving, but in reality, the digits are shifting to the left into larger place value slots. When you calculate $0.5 \times 10$, the $5$ moves from the tenths place to the ones place. Because our number system is 'Base-10,' every time a digit moves one spot to the left, it becomes 10 times more valuable. This is why mental math with powers of ten is so fast—you are simply reassigning the 'home' of each digit!