Question

reflect

1. when you move the decimal point, how can you know whether you are increasing or decreasing the number?
2. explain how the two steps of moving the decimal and multiplying by a power of 10 leave the value of the original number unchanged.

Explanation:

Question 1

Step 1: Understand decimal movement direction

If moving the decimal point to the right, the number of places moved is \( n \), the number is multiplied by \( 10^n \), so it increases. If moving left, multiplied by \( 10^{-n} \) (or divided by \( 10^n \)), so it decreases.

Step 2: Relate to number change

For example, take \( 5.6 \). Move decimal right 2 places: \( 560 = 5.6\times10^2 \) (increase). Move left 2 places: \( 0.056 = 5.6\times10^{-2} \) (decrease).

Step 1: Analyze decimal movement

Moving the decimal point \( n \) places is equivalent to multiplying (or dividing) by \( 10^n \) (or \( 10^{-n} \)).

Step 2: Analyze power of 10 multiplication

Multiplying by \( 10^k \) scales the number by \( 10^k \). To keep the value unchanged, the two steps must be inverses:

• If you move the decimal \( n \) places right (multiply by \( 10^n \)), you must multiply by \( 10^{-n} \) (or divide by \( 10^n \)) to cancel it.
• If you move the decimal \( n \) places left (multiply by \( 10^{-n} \)), you must multiply by \( 10^n \) to cancel it.

Mathematically: Let original number be \( x \). Move decimal \( n \) places right: \( x\times10^n \). Then multiply by \( 10^{-n} \): \( (x\times10^n)\times10^{-n} = x \). Similarly for left movement.

Answer:

• Moving decimal right: Number increases (multiplied by \( 10^n \), \( n \) = places moved).
• Moving decimal left: Number decreases (multiplied by \( 10^{-n} \), or divided by \( 10^n \)).

Question 2