Squared Numbers
In This Chapter
- Finding the value of a squared number that ends in 5
- Learning how to find the square of a number near 50
- Using the duplex to find the value of two-, three-, and four-digit numbers
- Figuring the sums of squares
Squared numbers are only too easy to type into a calculator, but if you need to do them in your head, it can be a bit tricky. However, there are ways you can find the values that allow you to go beyond the calculator and really understand squared numbers.
In this chapter, you learn processes for finding the value of squares and for adding squared numbers.
Finding Squares of Numbers Ending in 5
It’s very simple to find the square of numbers that end with 5. To get the first part of the answer, you apply the method called by one more than the one before. What does this mean? It means you simply find out what’s one more than the number before 5 and multiply those two numbers together. You then get the second part of your answer by squaring 5, or multiplying 5 by itself. Check out the following examples to see what I mean.
Example 1
Find the value of 352.
Step 1: To get the first part of the answer, apply the “by one more than the one before” method. What is one more than the first digit? One more than 3 is 4. Multiply the two numbers together.
3 × 4 = 12
Step 2: To get the second part of the answer, square 5.
5 × 5 = 25
Solution: The answer is 1,225.
Example 2
Find the value of 752.
Step 1: To get the first part, find out what’s one more than the first digit and multiply. One more than 7 is 8, so multiply them together.
7 × 8 = 56
Step 2: To get the second part of the answer, square 5.
5 × 5 = 25
Solution: The answer is 5,625.
SPEED BUMP
Keep in mind that this technique is only for finding out the squares of numbers that end with 5. Don’t try to use it with squares of other numbers!
Find the value of 1152.
Step 1: To get the first part, find out what’s one more than the first two digits and multiply. One more than 11 is 12, so multiply them together.
11 × 12 = 132
Step 2: To get the second part of the answer, square 5.
5 × 5 = 25
Solution: The answer is 13,225.
Finding Squares of Numbers Near 50
Finding the value of squared numbers near 50 is easy and fun. Like the method for numbers ending in 5, you only need to complete two steps to get the answer. Both steps involve using the excess or deficit—or how much more or less the number is than 50—to get the answer.
Numbers Above 50
If the number’s above 50, you get the first part by adding the excess to 25, and you get the second part by squaring the excess. Try this process out with the following examples.
Example 1
Find the value of 542.
Step 1: To get the first part of the answer, find out how much more the number is than 50 and add the excess to 25. In this case, 54 is 4 more than 50, so add that to 25.
25 + 4 = 29
Step 2: To get the second part of the answer, square the excess.
4 × 4 = 16
Solution: The answer is 2,916.
Find the value of 522.
Step 1: To get the first part, find out how much more the number is than 50 and add the excess to 25. In this case, 52 is 2 more than 50, so add that to 25.
25 + 2 = 27
Step 2: To get the second part, square the excess. The second part must be a double-digit number, so put a zero before this number.
2 × 2 = 4 or 04
Solution: The answer is 2,704.
Example 3
Find the value of 612.
Step 1: To get the first part, find out how much more the number is than 50 and add the excess to 25. In this case, 61 is 11 more than 50, so add that to 25.
25 + 11 = 36
Step 2: To get the second part, square the excess. Because you get a three-digit number, add the first 1 to 36. That makes the two parts of the answer 37 and 21.
11 × 11 = 121
1 + 36 = 37
Solution: The answer is 3,721.
Numbers Below 50
If the number’s below 50, instead of adding the difference to 25, you subtract it from 25. To get the second part of the answer, you do as you did before—square the difference between the number and 50. Let me show you how this works.
Find the value of 462.
Step 1: To get the first part of the answer, find out how much less the number is than 50 and subtract the deficit from 25. In this case, 46 is 4 less than 50, so subtract that from 25.
25 − 4 = 21
Step 2: To get the second part of the answer, square the deficit.
4 × 4 =16
Solution: The answer is 2,116.
Example 2
Find the value of 422.
Step 1: To get the first part, find out how much less the number is than 50 and subtract the deficit from 25. In this case, 42 is 8 less than 50, so subtract that from 25.
25 − 8 = 17
Step 2: To get the second part, square the deficit.
8 × 8 = 64
Solution: The answer is 1,764.
Example 3
Find the value of 382.
Step 1: To get the first part, find out how much less the number is than 50 and subtract the deficit from 25. In this case, 38 is 12 less than 50, so subtract that from 25.
25 − 12 = 13
Step 2: To get the second part, square the deficit. Because you get a three-digit number, add the first 1 to 13. That makes the two parts of the answer 14 and 44.
12 × 12 = 144
13 + 1 = 14
Solution: The answer is 1,444.
Using the Duplex to Find the Value of a Squared Number
To find out the squares of any given number, you have to learn a concept called the duplex. The duplex has a different meaning based on the number of digits of what’s being squared.
If the number being squared is just one digit, you simply use the formula a2, where a is the number. So, for example, the duplex of 7 is 49, because 72 = 49.
If the squared number has two digits, you use the formula 2ab, where a is the first digit and b is the second digit. Say you want to know the duplex of 81; you multiply the individual digits with 2 and get your answer: 2 × 8 × 1 = 16. So 81 has a duplex of 16. You can add variables based on the number of digits.
If the squared number has three digits, you go with the formula b2 + 2ac, where a is the first digit, b is the second digit, and c is the third digit. Say you want to find the duplex of 372. Here’s how you’d plug it in: 72 + (2 × 3 × 2) = 49 + 12 = 61. As you can see, the duplex of 372 is 61.
If the number being squared has four digits, you use the formula 2ad + 2bc. For example, here’s how you get the duplex of 7,351: (2 × 7 × 1) + (2 × 3 × 5) = 14 + 30 = 44.
But how does this apply to finding the value of squares? You do the duplex for every combination in a number and combine them to get the answer. The following sections show you how this is done.
Two-Digit Squares
To find the square of any two-digit number, you find three duplexes: one for the first digit (a2), one for the digit as a whole (2ab), and one for the second digit (a2). You then combine them to get your answer.
Find the value of 572.
Step 1: Find the duplexes for 5, 57, and 7 and put the three values together with slashes separating them; the slashes represent columns.
Duplex of 5: 52 = 25
Duplex of 57: 2 × 5 × 7 = 70
Duplex of 7: 72 = 49
25/70/49
Step 2: Add the three values from right to left. You can only have one digit in every column except the last, so put down 9 for this column and carry over the 4 to the next column.
Step 3: Add the carryover to 70: 70 + 4 = 74. Again, you can only have one digit in every column except the last, so put down 4 and carry over the 7 to the next column.
Step 4: Add the carryover to 25: 25 + 7 = 32. Put down 32.
Solution: The answer is 3,249.
Find the value of 742.
Step 1: Find the duplexes for 7, 74, and 4 and put the three values together with slashes separating them; the slashes represent columns.
Duplex of 7: 72 = 49
Duplex of 74: 2 × 7 × 4 = 56
Duplex of 4: 42 = 16
49/56/16
Step 2: Add the three values from right to left. You can only have one digit in every column except the last, so put down 6 and carry over the 1 to the next column.
Step 3: Add the carryover to 56: 56 + 1 = 57. Put down 7 and carry over the 5 to the next column.
Step 4: Add the carryover to 49: 49 + 5 = 54. Put down 54.
Solution: The answer is 5,476.
Example 3
Find the value of 862.
Step 1: Find the duplexes for 8, 6, and 86 and put the three values together with slashes separating them; the slashes represent columns.
Duplex of 86: 2 × 8 × 6 = 96
Duplex of 6: 62 = 36
64/96/36
Step 2: Add the values from right to left. Because you can only have one digit in every column except the last, put down 6 and carry over the 3 to the next column.
Step 3: Add the carryover to 96: 96 + 3 = 99. Put down 9 and carry over the other 9 to the next column.
Step 4: Add the carryover to 64: 64 + 9 = 73. Put down 73.
Solution: The answer is 7,396.
QUICK TIP
The duplex method for two-digit squares is an easy one to practice mentally. After you’ve learned it with pencil and paper, try picturing the problem in your head and the steps it takes to get the answer.
Three-Digit Squares
To find the square of any three-digit number, you need to find five duplexes: one for the first digit (a2), one for the first and second digit (2ab), one for the digit as a whole (b2 + 2ac), one for the second and third digit (2ab), and one for the third digit (a2). Like you did with the two-digit squares, you combine the duplex values to get your answer.
Find the value of 7462.
Step 1: Find the duplexes for 7, 74, 746, 46, and 6 and put the five values together with slashes separating them; the slashes represent columns.
Duplex of 7: 72 = 49
Duplex of 74: 2 × 7 × 4 = 56
Duplex of 746: 42 + (2 × 7 × 6) = 16 + 84 = 100
Duplex of 46: 2 × 4 × 6 = 48
Duplex of 6: 62 = 36
49/56/100/48/36
Step 2: Add the values from right to left. Because you can only have one digit in every column except the last, put down 6 and carry over the 3 to the next column.
Step 3: Add the carryover to 48: 48 + 3 = 51. Put down 1 and carry over the 5 to the next column.
Step 4: Add the carryover to 100: 100 + 5 = 105. Put down 5 and carry over the 10 to the next column.
Step 5: Add the carryover to 56: 56 + 10 = 66. Put down 6 and carry over the 6 to the last column.
Step 6: Add the carryover to 49: 49 + 6 = 55. Put down 55.
Solution: The answer is 556,516.
Example 2
Find the value of 3572.
Step 1: Find the duplexes for 3, 35, 357, 57, and 7 and put the five values together with slashes separating them; the slashes represent columns.
Duplex of 3: 32 = 9
Duplex of 35: 2 × 3 × 5 = 30
Duplex of 357: 52 + (2 × 3 × 7) = 25 + 42 = 67
Duplex of 57: 2 × 5 × 7 = 70
Duplex of 7: 72 = 49
9/30/67/70/49
Step 2: Add the values from right to left. Because you can only have one digit in every column except the last, put down 9 and carry over the 4 to the next column.
Step 3: Add the carryover to 70: 70 + 4 = 74. Put down 4 and carry over 7 to the next column.
Step 4: Add the carryover to 67: 67 + 7 = 74. Put down 4 and carry over the 7 to the next column.
Step 5: Add the carryover to 30: 30 + 7 = 37. Put down 7 and carry over the 3 to the next column.
Step 6: Add the carryover to 9: 9 + 3 = 12. Put down 12.
Solution: The answer is 127,449.
Example 3
Find the value of 6832.
Step 1: Find the duplexes for 6, 68, 683, 83, and 3 and put the five values together with slashes separating them; the slashes represent columns.
Duplex of 6: 62 = 36
Duplex of 68: 2 × 6 × 8 = 96
Duplex of 683: 82 + (2 × 6 × 3) = 64 + 36 = 100
Duplex of 83: 2 × 8 × 3 = 48
Duplex of 3: 32 = 9
36/96/100/48/9
Step 2: Add the values from right to left. Because you only have one digit in every column except the last, you can simply put down 9.
Step 3: In the next column, put down 8 and carry over the 4 to the next column.
Step 4: Add the carryover to 100: 100 + 4 = 104. Put down 4 and carry over the 10 to the next column.
Step 5: Add the carryover to 96: 96 + 10 = 106. Put down 6 and carry over the 10 to the next column.
Step 6: Add the carryover to 36: 36 + 10 = 46. Put down 46.
Solution: The answer is 466,489.
Four-Digit Squares
Finding the square of a four-digit number requires to first find seven duplexes: one for the first digit (a2); one for the first and second digit (2ab); one for the first, second, and third digit (b2 + 2ac); one for the digit as a whole (2ad + 2bc); one for the second, third, and fourth digit (b2 + 2ac); one for the third and fourth digit (2ab); and one for the fourth digit (a2). Like you did with the two-digit and three-digit squares, you combine the duplex values to get your answer.
SPEED BUMP
Remember, to get the correct answer, you must find the duplex for all of the combinations. So for four-digit numbers, you can’t simply do the formula for four digits and expect that to be the answer; you also need the values for each digit individually.
Find the value of 2,8942.
Step 1: Find the duplexes for 2; 28; 289; 2,894; 894; 94; and 4. Put the seven values together with slashes separating them; the slashes represent columns.
Duplex of 2: 22 = 4
Duplex of 28: 2 × 2 × 8 = 32
Duplex of 289: 82 + (2 × 2 × 9) = 64 + 36 = 100
Duplex of 2894: (2 × 2 × 4) + (2 × 8 × 9) = 16 + 144 = 160
Duplex of 894: 92 + (2 × 8 × 4) = 81 + 64 = 145
Duplex of 94: 2 × 9 × 4= 72
Duplex of 4: 42 = 16
4/32/100/160/145/72/16
Step 2: Add the values from right to left. Because you can only have one digit in every column except the last, put down 6 and carry over the 1 to the next column.
Step 3: Add the carryover to 72: 72 + 1 = 73. Put down 3 and carry over the 7 to the next column.
Step 4: Add the carryover to 145: 145 + 7 = 152. Put down 2 and carry over the 15 to the next column.
Step 5: Add the carryover to 160: 160 + 15 = 175. Put down 5 and carry over the 17 to the next column.
Step 6: Add the carryover to 100: 100 + 17 = 117. Put down 7 and carry over the 11 to the next column.
Step 7: Add the carryover to 32: 32 + 11 = 43. Put down 3 and carry over the 4.
Step 8: Add the carryover to 4: 4 + 4 = 8. Put down 8.
Solution: The answer is 8,375,236.
Example 2
Find the value of 1,2342.
Step 1: Find the duplexes for 1; 12; 123; 1,234; 234; 34; and 4. Put the seven values together with slashes separating them; the slashes represent columns.
Duplex of 1: 12 = 1
Duplex of 12: 2 × 1 × 2 = 4
Duplex of 123: 22 + (2 × 1 × 3) = 4 + 6 = 10
Duplex of 1234: (2 × 1 × 4) + (2 × 2 × 3) = 8 + 12 = 20
Duplex of 234: 32 + (2 × 2 × 4) = 9 + 16 = 25
Duplex of 34: 2 × 3 × 4 = 24
Duplex of 4: 42 = 161/4/10/20/25/24/16
Step 2: Add from right to left. Because you can only have one digit in every column except the last, put down 6 carry over the 1 to the next column.
Step 3: Add the carryover to 24: 24 + 1 = 25. Put down 5 and carry over the 2.
Step 4: Add the carryover to 25: 25 + 2 = 27. Put down 7 and carry over the 2 to the next column.
Step 5: Add the carryover to 20: 20 + 2 = 22. Put down 2 and carry over the 2 to the next column.
Step 6: Add the carryover to 10: 10 + 2 = 12. Put down 2 and carry over the 1 to the next column.
Step 7: Add the carryover to 4: 4 + 1 = 5. Put down 5; because it’s a single digit, you have no carryover.
Step 8: Because there’s no carryover to add in, simply put down 1.
Solution: The answer is 1,522,756.
Combined Operations: Sums of Squares
Now that you know how to find the value of individual numbers, let’s take it a step further by looking at how to add squared digits together. The sum of two-digit squares is a lot like finding the duplexes for two-digit numbers, except this time you’re adding the duplexes in reverse in groups. You first find the duplexes of the second digits and add them together (a2 + a2). You then find the duplexes for the numbers as a whole and add them (2ab + 2ab). You next find the duplexes of the first digits and add them together (a2 + a2).
The same is true for the sum of three-digit squares. You’ll be using modified versions of the formulas you used earlier in reverse to find the sums—one for the third digits (a2 + a2), one for the second and third digits (2ab + 2ab), one for the digits as a whole ([b2 + 2ac] + [b2 + 2ac]), one for the first and second digits (2ab), and one for the first digits (a2). Like with the two-digit squares, you combine the duplex values to get your answer.
You can probably guess how it goes for the sum of four-digit squares. You add the duplexes of the fourth digits (a2 + a2); the duplexes of the third and fourth digits (2ab + 2ab); the duplexes of the second, third, and fourth digits ([b2 + 2ac] + [b2 + 2ac]); the duplexes for the digits as a whole ([2ad + 2bc] + [2ad + 2bc]); the duplexes for the first, second, and third digits ([b2 + 2ac] + [b2 + 2ac]); the duplexes for the first and second digits (2ab + 2ab); and the duplex for the first digits (a2 + a2).
You can see how to do each type in the following examples.
Example 1
Solve the problem 412 + 332.
Step 1: Find the duplexes of the second digits and add. In this case, you add the duplexes for 1 and 3.
12 + 32 = 1 + 9 = 10
10
Step 2: Find the duplexes of the numbers as a whole and add. Here, add the duplexes of 41 and 33; then, add the carryover from the previous step.
(2 × 4 × 1) + (2 × 3 × 3) = 8 + 18 = 26
26 + 1 = 27
270
Step 3: Find the duplexes for the first digits and add. In this case, you add the duplexes of 4 and 3; then, add the carryover from the previous step.
42 + 32 = 16 + 9 = 25
25 + 2 = 27
2770
Solution: The answer is 2,770.
Example 2
Solve the problem 2032 + 1122 + 4222.
Step 1: Find the duplexes of the third digits and add. In this case, you add the duplexes of 3, 2, and 2.
32 + 22 + 22 = 9 + 4 + 4 = 17
17
Step 2: Find the duplexes of the second and third digits and add. Here, add the duplexes of 03, 12, and 22; then, add the carryover.
(2 × 0 × 3) + (2 × 1 × 2) + (2 × 2 × 2) = 0 + 4 + 8 = 12
12 + 1 = 13
137
Step 3: Find the duplexes of the digits as a whole and add. In this case, you add the duplexes of 203, 112, and 422; then, add the carryover.
(02 + [2 × 2 × 3]) + (12 + [2 × 1 × 2]) + (22+ [2 × 4 × 2]) = 12 + 5 + 20 = 37
37 + 1 = 38
3837
Step 4: Find the duplexes of the first and second digits and add. Here, add the duplexes of 20, 11, and 42; then, add the carryover.
(2 × 2 × 0) + (2 × 1 × 1) + (2 × 4 × 2) = 0 + 2 + 16 = 18
18 + 3 = 21
21837
Step 5: Find the duplexes of the first digits and add. In this case, add the duplexes of 2, 1, and 4; then, add the carryover.
22 + 12 + 42 = 4 + 1 + 16 = 21
21 + 2 = 23
231837
Solution: The answer is 231,837.
QUICK TIP
If you encounter a problem in which the numbers being added don’t have the same number of digits, add zeroes. For example, for the problem 3412 + 82 + 212, change it to 3412 + 0082 + 0212. Adding the zeroes still gives you the same answer while also making it less complicated to use the duplex formulas.
Example 3
Solve the problem 1,2562 + 4,7652 + 3,8232 + 5,9952.
Step 1: Find the duplexes of the fourth digits and add. Here, add the duplexes of 6, 5, 3, and 5.
62 + 52 + 32 + 52 = 36 + 25 + 9 + 25 = 95
95
Step 2: Find the duplexes of the third and fourth digits and add. In this case, add the duplexes of 56, 65, 23, and 95; then, add the carryover.
(2 × 5 × 6) + (2 × 6 × 5) + (2 × 2 × 3) + (2 × 9 × 5) = 60 + 60 + 12 + 90 = 222
222 + 9 = 231
2315
Step 3: Find the duplexes of the second, third, and fourth digits and add. Here, add the duplexes of 256, 765, 823, and 995; then, add the carryover.
(52 + [2 × 2 × 6]) + (62 + [2 × 7 × 5]) + (22 + [2 × 8 × 3]) + (92 + [2 × 9 × 5]) = 49 + 106 + 52 + 171 = 378
378 + 23 = 401
40115
Step 4: Find the duplexes of the numbers as a whole and add. In this case, add the duplexes of 1,256; 4,765; 3,823; and 5,995; then, you add the carryover.
([2 × 1 × 6] + [2 × 2 × 5]) + ([2 × 4 × 5] + [2 × 7 × 6]) + ([2 × 3 × 3] + [2 × 8 × 2]) + ([2 × 5 × 5] + [2 × 9 × 9]) = 32 + 124 + 50 + 212 = 418
418 + 40 = 458
458115
Step 5: Find the duplexes of the first, second, and third digits and add. Here, add the duplexes of 125, 476, 382, and 599; then, add the carryover.
(22 + [2 × 1 × 5]) + (72 + [2 × 4 × 6]) + (82 + [2 × 3 × 2]) + (92 + [2 × 5 × 9]) = 14 + 97 + 76 + 171 = 358
358 + 45 = 403
4038115
Step 6: Find the duplexes of the first and second digits and add. Here, add the duplexes of 12, 47, 38, and 59; then, add the carryover.
(2 × 1 × 2) + (2 × 4 × 7) + (2 × 3 × 8) + (2 × 5 × 9) = 4 + 56 + 48 + 90 = 198
198 + 40 = 238
23838115
Step 7: Find the duplex of the first digits and add. In this case, add the duplexes of 1, 4, 3, and 5; then, add the carryover.
12 + 42 + 32 + 52 = 1 + 16 + 9 + 25 = 51
51 + 23 = 74
74838115
Solution: The answer is 74,838,115.
- For squared numbers ending in 5, use the “by one more than the one before” to get the first part of your answer.
- If you have a squared number near 50, you can find the answer by using the excess or deficit.
- The formulas a2, 2ab, b2 + 2ac, and 2ad + 2bc give you the duplex of one digit, two digits, three digits, and four digits respectively.
- When adding squared numbers, you simply group by duplex and add.