6-2=4-2=2-2=0
6÷2=3
9-2=7-2=5-2=3-2=1
Subtracting by 2 again would go past zero.
9÷2=4 with a remainder of 1 or 9=2×4+1
What happens when we divide by negative numbers? We divide just as we would with positive numbers, but we have to remember our sign rules for multiplication. Remember that an odd number of negative signs in multiplication gives a negative number. So if we multiply two negative numbers, we get a positive product. If we multiply two positive numbers we get a positive product. So if we divide a positive number, we must have gotten it by multiplying two numbers with the same sign. That means, in division, the divisor and the quotient must have the same sign. If the dividend starts negative, then the divisor and quotient must have different signs.
8÷-2=-4
The dividend is positive, the divisor gives the sign of the quotient.
-8÷-2=4
-8÷2=-4
The dividend is negative, the quotient has the opposite sign of the divisor.
9÷-2=-4 with a remainder of 1 because 9=-2×-4 + 1
-9÷-2=4 with a remainder of -1 because -9=-2×4 - 1
Let's look at division of large numbers.
12345÷567
Just start subtracting by 567 and keep count.
12345-567=11778 (1st division)
11778-567=11211 (2nd division)
11211-567=10644 (3rd division)
10644-567=10077 (4th division)
10077-567=9510 (5th division)
9510-567=8943 (6th division)
8943-567=8376 (7th division)
8376-567=7809 (8th division)
7809-567=7242 (9th division)
7242-567=6675 (10th division)
6675-567=6108 (11th division)
6108-567=5541 (12th division)
5541-567=4974 (13th division)
4974-567=4407 (14th division)
4407-567=3840 (15th division)
3840-567=3273 (16th division)
3273-567=2706 (17th division)
2706-567=2139 (18th division)
2139-567=1572 (19th division)
1572-567=1005 (20th division)
1005-567=438 (21st division)
So 12345÷567=21 with a remainder of 438
Doing division by multiple subtractions can be tedious and presents lots of opportunites for errors. We start by looking only at the number in the highest positions. So we want to divide 10000 by 500. Since there are two zeros on the right of 5, we can eliminate two zeros from the right in both numbers. So now we are looking at 100÷5. From our multiplication table, we don't have a multiple of 5 that equals 100. But we do have a multiple of 10 that equals 100 and a multiple of 5 that equals 10. 5×2×10=100. Using the associativity of the integers, we have 5×(2×10)=5×20=100. Putting the zeros back in, we have that 10000÷500=20.
Start by multiplying 567 by 20
| 5 | 6 | 7 | ||
| × | 2 | 0 | ||
| 1 | 1 | 3 | 4 | 0 |
12345-11340=1005
1005 > 567 so multiply by 21 instead.
| 5 | 6 | 7 | ||
| × | 2 | 1 | ||
| 1 | 1 | 9 | 0 | 7 |
12345-11907=438
438 < 567
So 12345÷567=21 with a remainder of 438
This is essentially how the Indians developed long division. Their notation is different, though. We're going to use the ) notation of long division with the divisor to the left and the dividend to the right. We are going to write our quotient above the dividend, carfeully observing positioning. The result at the bottom will be the remainder. We start like this:
| — | — | — | — | — | ||||
| 5 | 6 | 7 | ) | 1 | 2 | 3 | 4 | 5 |
Now we count off positions in the dividend until we get a number that is larger than the divisor. 1234 is larger than 567. How many times does 5 go into 12? 2 times. We place 2 above the 4 in 1234. We will multiply 567 by 2 and place that product underneath 1234. Then we subtract that from 1234 and place the difference underneath. It looks like this.
| 2 | ||||||||
| — | — | — | — | — | ||||
| 5 | 6 | 7 | ) | 1 | 2 | 3 | 4 | 5 |
| - | 1 | 1 | 3 | 4 | ||||
| — | — | — | — | |||||
| 1 | 0 | 0 |
Now we bring down the 5 in the next column of 12345 to make a number greater than 567. How many times does 5 go into 10? 2 times. Place the 2 above the 5. Multiply 567 by 2 and place the product underneath 1005.
| 2 | 2 | |||||||
| — | — | — | — | — | ||||
| 5 | 6 | 7 | ) | 1 | 2 | 3 | 4 | 5 |
| - | 1 | 1 | 3 | 4 | ||||
| — | — | — | — | |||||
| 1 | 0 | 0 | 5 | |||||
| - | 1 | 1 | 3 | 4 | ||||
| — | — | — | — |
1134 is greater than 1005, so we reduce the multiplier by 1. Place 1 above the 5 and multiply 567 by 1. Place 567 underneath 1005 and subtract.
| 2 | 1 | |||||||
| — | — | — | — | — | ||||
| 5 | 6 | 7 | ) | 1 | 2 | 3 | 4 | 5 |
| - | 1 | 1 | 3 | 4 | ||||
| — | — | — | — | |||||
| 1 | 0 | 0 | 5 | |||||
| - | 5 | 6 | 7 | |||||
| — | — | — | — | |||||
| 4 | 3 | 8 |
438 < 567 so we're done.
12345=567×21+438
Division is neither associative nor commutative. The grouping and the order of the numbers matters.
(18÷6)÷3 = 3÷3 = 1
18÷(6÷3) = 18÷2 = 9
Finally, there is a special number that we look at regarding division. It will become very important in the next blog. It is the greatest common divisor. The greatest common divisor, or gcd, is the largest number that evenly divides two numbers. We look at the prime number divisors of a two numbers and collect the divisors they have in common.
You get the prime factors of a number by dividing it by prime numbers until you get to 1. If the number is even then you can divide by 2, for example, 246 is even and 246÷2=123. If a number ends in 5 then you can divide dy 5, for example, 85 ends in 5 and 85÷5=17. If the sum of the digits of a number is a multiple of 3 then the number can be divided by 3, for example, in 123 the digits add up to 6 which is a multiple of 3 and 123÷3=41. Here's the url in Wikipedia that lists the divisibility rules. http://en.wikipedia.org/wiki/Divisibility_rule
The prime factors of 8 are 2,2 and 2. The prime factors of 12 are 2,2 and 3. Both 8 and 12 have a pair of 2's in common. So the greatest common divisor of 8 and 12 is 2×2=4. Another way to find the gcd is to multiply the two numbers together and divide by the lcm. You can similarly find the lcm by multiplying the two numbers together and divide by the gcd. The product of 8 and 12 is 96. The lcm of 8 and 12 is 24. 96 divided by 24 is 4.
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