# Magic of Nines with Digital Roots

Remember, that the digital root of a number is the number obtained by adding all the digits, then adding the digits of that sum, and then continuing until a single-digit number is reached. Zero can never be a digital root only 1,2,3,4,5,6,7,8 and 9.

Digital roots were known to the Roman bishop Hippolytos as early as the third century. It was employed by Twelfth-century Hindu mathematicians as a method of checking answers to multiplication, division, addition and subtraction and we proved that it works this week.

For example, the digital root of 65,536 is 7, because 6 + 5 + 5 + 3 + 6 = 25 and 2 + 5 = 7.

USING DIGITAL ROOTS TO VERIFY ADDITION

Suppose we are adding two numbers:

345 + 789.

• The Digital Root of 789 is 6.
• The Digital Root of 345 is 3
• Now add 3 + 6 = 9
• This implies that the Digital Root of our final answer
• obtained after adding 345 + 789 will be 9
• Let’s verify that 345 + 789 = 1134
• The Digital Root of 1134=1+1+3+4=9 This equals the sum obtained at by adding the digital roots of the original two numbers above.

USING DIGITAL ROOTS TO VERIFY SUBTRACTION

Suppose we are subtracting two numbers:

789 -- 345

• The Digital Root of 789 is 6.
• The Digital Root of 345 is 3
• Now subtract 6 -- 3 = 3
• This implies that the Digital Root of our final answer                                                                      obtained after subtracting 345 from 789 will be 3.
• Let’s verify that 789 -- 345 = 444
• The Digital Root of 444=4+4+4=1+2=3 This equals the difference obtained by subtracting the digital roots of the original two numbers above. BEWARE: if your top digital root is less than the bottom, before you subtract, add 9 to the top digital root.

USING DIGITAL ROOTS TO VERIFY MULTIPLICATION

Suppose we are multiplying two numbers:  12 x 8

• The Digital Root of 12 is 3.
• The Digital Root of 8 is 8.
• Now multiply 3x8=24; whose digital root is 6.
• This implies that the Digital Root of our final answer
• obtained after multiplying 12 by 8 will be 6
• Let’s verify that 12x8=96
• The Digital Root of 96 is 9+6=15 and 1+5=6 This equals the product obtained at by multiplying the digital roots of the original two numbers above.

Now the real fun begins. These appear to be magic tricks, but the mathematician knows it is just pure digital roots and nines.

MAGIC MATH WITH SUBTRACTION AND DIGITAL ROOTS

1. Each student writes down any-digit number. For example, 81298.

2. Each student writes down a second same digit number by rearranging the order of the digits of

their first number. For example, 21889.

3. Each student subtracts their smaller number from their larger number. For example, 81298 - 21889 = 59409.

4. The students do not know it, but the digital root of everyone's answer is 9. You will use this fact to do the trick.

5. Each student circles one of the digits from their answer. For example, 59409.

(This trick is much easier to do if they do not cross out a zero. The reason will become clear once

you know how to do the trick.)

6. The student tells you the numbers that are left in their answer. For example, 5, 9, 0 and 9.

7. Calculate the digital root of the numbers they tell you. For example, 5 + 9 + 0 + 9 = 23 which

becomes 2 + 3 = 5.

8. You know the digital root of their complete answer is 9 and you know the digital root of the

numbers they gave to you is 5, so you know they must have circled a 4 (just subtract the digital root of the numbers they gave you from 9 and that is the number they circled). If the student had crossed out a 9, they would have told you 5, 4, 0 and 9. The digital root of those numbers is 9. This tells you that they either crossed out a zero (9 + 0 = 9) or a nine (9 + 9 = 18 which becomes 1 + 8 = 9). Since it is not possible to distinguish a missing zero from a missing nine, you have to prevent the students from crossing out one of them. Say something like "Cross out any number you like from your answer, except for zero, because zeros are too easy.”

An example is worked out on page 7 of the pdf. Here are some simple steps to remember when doing this with another person (JUST REMEMBER THAT YOUR TEST SUBJECT MUST BE ABLE TO DO SUBTRACTION WITH BORROWING).

MAGIC MATH WITH SUBTRACTION AND DIGITAL ROOTS

1. Choose an n-digit number (THAT IS A NUMBER WITH ANY NUMBER OF DIGITS) and write it down (different digits). This can be done with any number of digits, but I chose 4 for simplicity during the class.

2. Jumble up the digits and write the smaller of the two numbers underneath the larger number.

3. Now subtract the smaller number from the bigger number and don't tell me the answer.

4. Circle one of the non-zero digits in your answer and tell me through Chat, the remaining digits, in any order.

5. I will tell you the number you circled.

Magic of Nines: MAGIC OF THREE DIGIT NUMBERS

1.  Have someone take any three digit number as long as the numbers; do not repeat the digits.

2. Reverse the order of that number;

3. Subtract the smaller number from the larger;

4. Have them tell you the first digit of the answer;

5. There can only be nine possible answers (all have a 9 in the middle and the first and third digits have a sum of nine); they are:

• 099
• 198
• 297
• 396
• 495
• 594
• 697
• 798
• 891

So, they tell you that their answer starts with a 2, you know their answer must be 297 because all of the answers have a 9 in the middle and the

sums of the first and last digits will always be 9.

An example is worked out on page 9 of the pdf. Here are some simple steps to remember when doing this with another person (JUST REMEMBER THAT YOUR TEST SUBJECT MUST BE ABLE TO DO SUBTRACTION WITH BORROWING).

Magic of Nines: MAGIC OF THREE DIGIT NUMBERS

1.  Write down any three digit number as long as the numbers do not repeat the digits.

2.  Reverse the digits.

3.  Subtract the smaller number from the larger number.

4.  Tell me the first or last digit of the answer and you have to tell me whether it is the first or the last digit.

During class, one of the students noticed the only possible answers are all multiples of 99. When I asked them what other factors are common among all of the possible answers, they came up with some interesting answers. The common factors of 99,198,297,396,495,594,693,792, and 891

are 1, 3, 9, 11, 33, and 99. Some Mathletes believed that 27 should be a common factor because 3 x 9 = 27, but we can see that none of these numbers has prime factors of 3x3x3=27. This is a logical conjecture by the Mathletes since 3x11=33 and 3x33=99 so why not 3x9?

Magic of Nines: ADDING NUMBERS WITH ANTI-NINES

1. Have someone pick a large number of any number of digits.

2. Have them write down a second number of the same number of digits.

3. Have them write down a third number of the same number of digits.

4. You write two anti-nines for the first two numbers (the sum of each digit
and its anti-nine is nine).

5. Now, here is the magic, the sum of all five numbers is the third number:

(a) take away two, and

(b) put the number 2 in front of that number. An example is worked out on pages 11 and 129 of the pdf. Here are some simple steps to remember when doing this with another person (JUST REMEMBER THAT YOUR TEST SUBJECT MUST BE ABLE TO DO ADDITION WITH CARRYING).

Magic of Nines: ADDING NUMBERS WITH ANTI-NINES

1. Pick a large number of any number of digits. Let's use 5 different digits.

2. Write down a second number of the same number of different digits.

3. Now, write down a third number of the same number of digits.

4. I will now write down two more numbers with the same number of digits and add the five numbers in a few seconds.

Your job this week is to practice each of these Magic of Nines activities so you can show your family, friends, and teachers when you get back to school.

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