Lesson 32: Expanded Notation

New Concept

Expanded notation breaks a number into a sum, revealing the true value of each digit based on its place in the number.

To write a number in expanded notation, we write each nonzero digit times its place value.

For instance, the number 8,430 is written as:

What’s next

This is your starting point. Next, you'll walk through examples converting numbers between standard and expanded forms and apply this skill to compare values.

Property

To write a number in expanded notation, we write each nonzero digit times its place value.

Examples

$27,000 = (2 \times 10,000) + (7 \times 1000)$
$1760 = (1 \times 1000) + (7 \times 100) + (6 \times 10)$
$8050 = (8 \times 1000) + (5 \times 10)$

Explanation

Think of this as showing a number's secret identity! You break the number down by multiplying each digit (except zero) by its place value, like thousands or hundreds. It's like writing out the full recipe for the number, showing all the valuable ingredients that add up to the final tasty total. We ignore zeros because they add no value.

Property

Standard notation is our usual way of writing numbers, created by adding the values from expanded notation together.

Examples

$(5 \times 1000) + (2 \times 100) + (8 \times 10) = 5280$
$(6 \times 1000) + (4 \times 100) = 6400$
$(7 \times 100) + (5 \times 1) = 705$

Explanation

This is like putting a super-team back together! You take the expanded parts, like $(5 \times 1000)$ and $(8 \times 10)$, and combine their powers by adding them up. The result is the single, everyday number we all recognize. It’s the opposite of expanding—you’re assembling the final hero from its individual power sources.

Property

When we calculate the amount of time between two events, we are calculating elapsed time (the amount of time that has passed).

Examples

To find the time from 7:15 a.m. to 11:10 a.m., calculate $10:70 - 7:15 = 3 \text{ hours } 55 \text{ minutes}$.
What time is $2\frac{1}{2}$ hours after 10:43 a.m.? $10:43 + 2:30 = 12:73$, which becomes 1:13 p.m.
How long is it from 2:10 p.m. to 4:45 p.m.? $4:45 - 2:10 = 2 \text{ hours } 35 \text{ minutes}$.

Explanation

You're a time detective solving a mystery! To find out how long something took, subtract the start time from the end time. If the end minutes are smaller than the start minutes, you must borrow! 'Steal' one hour from the end time and turn it into 60 extra minutes. Now you have enough to solve the problem!