Let's find the square of 432 using the diagonal method. Here's a step-by-step breakdown:
1. Write the number in a grid form: The number 432 has 3 digits, so we'll create a 3x3 grid. Each cell will hold the product of a pair of digits from the number.
```
4 | 3 | 2
____|_____|____
4 | | |
---|----|-----|-----
3 | | |
---|----|-----|-----
2 | | |
```
2. Fill in the grid with products of digits: Multiply each pair of digits and place the product in the appropriate cell.
```
4 | 3 | 2
____|_____|____
4 | 16 | 12 | 8
---|----|-----|-----
3 | 12 | 9 | 6
---|----|-----|-----
2 | 8 | 6 | 4
```
3. Draw diagonals from the bottom-left to top-right corners.
```
4 | 3 | 2
____|_____|____
4 | 16 | 12 | 8
---|----|-----|-----
3 | 12 | 9 | 6
---|----|-----|-----
2 | 8 | 6 | 4
Diagonals:
- From bottom-right: 4
- Next diagonal: 6, 8
- Next diagonal: 6, 9, 8
- Next diagonal: 12, 12, 4
- From top-left: 16
```
4. Add the numbers along each diagonal:
- First diagonal: 4 = 4
- Second diagonal: 6 + 8 = 14
- Third diagonal: 6 + 9 + 8 = 23
- Fourth diagonal: 12 + 12 + 4 = 28
- Fifth diagonal: 16 = 16
So, we have:
- [4, 14, 23, 28, 16]
5. Adjust for carries in each diagonal sum:
- First diagonal: 4 (no carry)
- Second diagonal: 14 (4, carry 1) → Result: 4, next carry: 1
- Third diagonal: 23 (23 + 1 from carry = 24, 4, carry 2) → Result: 4, next carry: 2
- Fourth diagonal: 28 (28 + 2 from carry = 30, 0, carry 3) → Result: 0, next carry: 3
- Fifth diagonal: 16 (16 + 3 from carry = 19, 9, carry 1) → Result: 9, next carry: 1
6. Account for the final carry: The carry from the fifth diagonal is 1.
7. Write out the final number by combining the results from each diagonal sum:
- Final number: 1 (from the carry) combined with diagonals → 90464
Therefore, the square of 432 is 90464 using the diagonal method.
