Key Concepts in Base Ten
1. Place Value
Each digit in a number has a value based on its position:
| Place | Value | Example (Number: 347) |
|---|---|---|
| Hundreds | 100 | 3 (3 × 100 = 300) |
| Tens | 10 | 4 (4 × 10 = 40) |
| Ones | 1 | 7 (7 × 1 = 7) |
- 347 = 300 + 40 + 7
2. Bundling and Regrouping
* Bundling – to form a new grouping of 10 items from existing items. For example, to form 1 ten (10) from 10 ones (1,2,3,4,5,6,7,8,9,10) from existing items.
* Regrouping – The process of changing the positions of the numbers in the calculation to get a result when adding or subtracting (e.g. in addition you carry the sum of all the digits together until the sum is less than 10 or subtracting, you borrow from the next higher number).
3. Comparing Numbers
- You can use > (greater than), < (less than), or = (equal to) to compare numbers.
- You have to compare digits from left to right.
Example:
- 253 vs. 248 → Compare hundreds (2 = 2), then tens (5 > 4), so 253 > 248.
Operations in Base Ten
1. Addition with Regrouping
Steps
1. Starting with the ones place, add the numbers in the ones column. If the result is greater than or equal to 10, regroup.
2. Next, add the number in the tens column to the number you just regrounded. Also, if there was a carry over from the ones (1 being carried from the first column, if necessary), you must include that value in this calculation. Note, if you do not need to carry the value(s) over to the next column, you can use them to calculate the total (see last example below).
3. Lastly, repeat with the numbers in the hundreds, etc. column.
Example
148+76=224
1. 3rd Column (The Column that is the Largest and Represents the Greatest Value) is the First Column to be Added together – The 100’s value 1+1=2.
2. 2nd Column Numbers should be Added Together (which represents tens) 4+7=11. Add the carryover value from column 1 to column 2 =12.
3. 3rd Column Numbers should be Added Together (which represents units [on] 8+6= 14. 1 (carryover) added to 4=5.
Final total =224.
2. Subtraction with Regrouping
Steps:
- Subtract ones. If the top digit is smaller, borrow from the tens.
- Repeat for tens, hundreds, etc.
Example:
- 324 – 157:
- Ones: 4 < 7 → Borrow 1 ten (14 – 7 = 7).
- Tens: 1 (left after borrowing) – 5 → Borrow 1 hundred (11 – 5 = 6).
- Hundreds: 2 – 1 = 1.
- Total: 167
3. Multiplying by 10, 100, etc.
- To multiply numbers by ten, one hundred, or similar, shift each digit toward the left. Automatic zeros will be added to the place value. The following examples are used to illustrate this process:
Examples:
- 6 × 10 = 60
- 23 × 100 = 2,300
4. Dividing by 10, 100, etc.
- Shift digits right (remove a zero for ÷10, two zeros for ÷100, etc.).
Examples:
- 50 ÷ 10 = 5
- 3,000 ÷ 100 = 30
Real-World Applications
Every day, we use this information; our money uses base ten values ($1.25 = 1 dollar plus 2 dimes plus 5 pennies), all of our measurements are based on metric systems as shown above (1 meter = 100 centimeters), and we also use base ten values for hour-to-minute conversions (60 seconds = 1 minute); however, we do not use base ten values when determining conversions between hours and minutes.