Advice 1: How to divide in the mind

In connection with the development of technology has eliminated the need to produce in the mind of the mathematical calculations. However, division without a calculator, computer and paper, pencil – it's good exercise for the brain and confidence in the event of unforeseen circumstances.
How to divide in the mind
Instruction
1
A few decades ago in the regular education schools there was a subject "Oral account". The children were taught to produce in the mind of the basic mathematical operations: addition, subtraction, multiplication and division, which can be considered the most difficult of them.
2
The division implies a quick search of the maximum of the divider. Method of oral division requires knowledge of techniques reduced the school division and multiplication tables. In addition, you need to train memory to learn to keep in mind all intermediate calculations, especially if the numbers are large.
3
The expansion of the private for sostavlenie, you need to divide the number by 7 3647. Imagine private as the sum of the numbers 3500 and 147. In this example, the 3500 is the biggest obvious the number, the smaller the source, which is divided by 7 without residue:3647/7 = 3500/7 + 147/7 = 500 + 147/7 = 500 + 21 = 521.
4
Division "column," in the mind, as in destinyland imagine a sheet of paper and imaginary pencil and make the calculations. This method requires a good visual memory, which, however, it is possible to train a regular exercise program in the account. This method is preferred by many because it is familiar from school days, though not as fast as the previous one.
5
Division by 10, 100, 1000, etc. This method involves the separation of a corresponding number of commas, starting with the right side of the number. For example, divide the number 567890 10,000:567890/10000 = 56,7890 – branch four zeros.
6
Division by 0.1, 0.01, etc., This scenario implies a multiplication of 1 with a corresponding number of subsequent zeros, i.e., a decimal fraction upside down. For example, divide the number 78,765 on 0,0001:78,765/0,0001 = 78,765*10000 = 787650.
7
The division into decimal will kobetamendi her mentally in common, for example, 0.5 for 1/2. Multiply the original number by the denominator and divide by the numerator. For example, divide the number 2250 for 0,75:2250/0,75 = 2250/(3/4) = 2250*4/3 = 9000/3 = 3000.
8
Dividing by 5, 50, 500, etc. Replace the divisor by the appropriate fraction: 5 = 10/2; 50 = 100/2, etc. Now it is enough to separate the private two decimal places and multiply by 2. For example, divide 1750 on 50:1750/50 = 1750*2/100 = 3500/100 = 35.
9
Similarly, there is a division by 2, 5, 25, etc.: the divisor is replaced by the appropriate fraction with 4 in the denominator. Of 1.25, 12.5 and so on the fraction with 8 in the denominator:285/2,5 = 285*4/10 = 1140/10 = 114;600/12,5 = 600*8/100 = 4800/100 = 48.

Advice 2: How to divide number by fraction

A fraction is a noninteger, or augmented the number, such as 1/2 (=0.5) or 7,5/5 (=1.5 m). Sometimes a fraction can be any integer numberm, for example, 20/5 (=4), but its entry does not have the mathematical meaning, which is entered in the roll.
How to divide number by fraction
Instruction
1
First recall that the simple or ordinary fraction can be written as X / Y where X is the numerator and Y the denominator. For example, 1/4, or 0.25 in digital recording. For convenience of further calculations, it is recommended to write the fraction vertically: a numerator, a horizontal bar dividing under him, and the denominator in the runway.To divide a number by a fraction, you need to present the number as a fraction. Since the number is the number of integer parts, then it goes in the denominator and in the numerator of the prescribed what is the number of divided parts to obtain the very same himself – that is, one. 8 should be written as 8/1, and 263 – how 263/1, and so on.
2
After that you need to divide number to fraction. Suppose you have the number 127 and the fraction 4/15. Then the operation 127 : 4/15 should be written as follows:127/1 : 4/15;
3
It turns out the three-story fraction at which the average division (division of fractions) should be replaced by multiplication, and the numerator and denominator flip:127/1 * 15/4;
4
Writing this action in the usual fraction with a horizontal division, you will receive:(127*15)/4;the Result of 467 1/4.
5
Counting on the calculator every roll, you will receive the following:127 : 1 = 127
4 : 15 = 0,2666...
127 : 0,2666... = 476, 476 2500001 or 1/4.The results coincide.

Advice 3: How to learn to do long division

The process of division in a column is the sequential execution of elementary arithmetic operations. To learn how to divide in a column, you just need to practice it a few times. The algorithm of dividing in a column let us look at the following examples - divide in a column of integers with no remainder, remainder and fractional numbers represented in decimal.
How to learn to do long division
You will need
  • - pen or pencil
  • - a sheet of paper in a cage.
Instruction
1
Division without a remainder. Divide 1265 55.
Swipe down short vertical line with a height of several cells. From this line a perpendicular swipe to the right. It turned out the letter "T" piled on the left side. Above the horizontal part of the sunken letter "T" spell a divider (55) and to its left in the same place, behind the vertical part of the T – divisible (1265). Usually, the dividend is first written, and then put a division sign in the column (filled up one side with the letter "T"), and after the divider.
2
Determine what portion of the dividend (counting from left to right the highest level) is divided by the divisor. That is: 1 55 no 12 55 – no, 126, 55, Yes. The number 126 is called incomplete severable.
3
Think about the mind, for what number N you need to multiply the divisor to get the number equal to or as close as possible (but no more) to the magnitude of the partial dividend. That is: 1*55 – is not enough, 3*55=165 – a bit too much. So, our choice is number 2. Write it under the divisor (below the horizontal part of the sunken letter "T").
4
Multiply 2 by 55 and write down the resulting number is 110 strictly under incomplete digits of the dividend left – to-right: 1 under 1, 1 under 2 and under 0 6. Top 126, bottom 110. Spend under 110 short horizontal line.
5
Subtract 126 from the number 110. Received 16. Clearly write down the numbers one below the other under the drawing line. That is, from left to right: number 1 number 110 is empty, the numeral 1 – 1, and the numeral 0 – 6. The number 16 is the residual, which must be less than the divisor. If it was bigger divisor, the integer N was chosen properly – you need to increase it and repeat the previous steps.
6
Carry to the next digit of the dividend (figure 5) and record it to the right of the number 16. It turned out 165.
7
Repeat the third step for the relationship 165 to 55, that is, find the number Q by multiplying the divisor by which the number of turns as close as possible to 165 (but not more). This 3 – 165 is divided into 55 without a trace. Write down the figure 3 the right of the number 2 below the line, held under the divider. This is the answer: a private relationship 1265 55 is 23.
8
The modulo operator. Divide 1276 55.Repeat all the same steps as in the division without a remainder. The number N is still 2, but the difference between the 127 and 110 is equal to 17. We carry 6 and determine the number of Q. It is also still equal to 3, but now there is the rest: 176 – 165 = 11. The remainder 11 is less than 55, everything seems to be fine. But to demolish something and nothing more...
9
Add the dividend to the right of the zero and put a comma after the number 3 in private (the number that can be obtained through division, and is recorded below the line, held under the divider).
10
Carry added in divisible zero (write it on the right of 11) and check if there is a possibility to divide the resulting number by the divisor. The answer is Yes: 2 (let us denote it as the number of (G) multiplied by 55 is 110. The answer is of 23.2.If carried in the previous step zero would not be enough to balance with the finished zero was more of the divider would need to add another zero in divisible and put 0 in private after the decimal point (would 23,0...).
11
Division in column of decimal fractions.Move the decimal point the same number of digits to the right in the divisor and divisible so that there were integers. Further, the division algorithm is the same.
Note
Write down all the numbers strictly under each other according to the stated recommendations is not allowed to make a mistake during the calculations.

Advice 4: Why can't you divide by zero

Divide by zero is impossible, every schoolboy knows that, but to many it is unclear why. The reasons for this rule can be found only at the high school, and then only if you study math. In fact, the basis of the fact that zero cannot be split, and not so complicated. To find out it would be very interesting to many students.
Why can't you divide by zero
The reason that you can't divide by zero, lies in mathematics. While in arithmetic there are four basic operations on numbers (this is addition, subtraction, multiplication, and division) in mathematics those are only two of them (addition and multiplication). They are included in the definition of a number. To determine what is the subtraction and division, you need to use addition and multiplication, and to breed a new operation from them. To understand this point, it is useful to consider some examples. For example, an operation 10-5, with the perspective of a student of the school, means that the number 10 is subtracted, the number 5. But the math would answer the question about what's going on here, otherwise. This operation would be reduced to the equation x+5=10. The unknown in this problem is x, it is the result of the so-called subtraction. The division is is similar. It's only similarly expressed through multiplication. In this case, the result is just the right number. For example, 10:5 a mathematician would write as 5*x=10. This problem has a unique solution. Considering all this, it is possible to understand why you cannot divide by zero. Entry 10:0 would become 0*x=10. That is, the result would be a number that when multiplied by 0 gives you another number. But we all know the rule that any number multiplied by zero, gives zero. This property is included in the concept of what is zero. So it turns out that the problem of how to divide a number by zerohas no solution. This is a normal situation, many problems in mathematics have no solution. But as it may seem, this rule has one exception. Yes, no number can not divide by zero, but zero is possible? For example, 0*x=0. It's true equality. But the problem is that in the place of x can be absolutely any number. Therefore, the result of this equation would be a perfect uncertainty. There is no reason to prefer any one outcome. So zero to zero to share too. However, in the mathematical analysis of such uncertainties can handle. Check to see if there in the problem of additional conditions, which "reveal uncertainty" as it's called. But in arithmetic do not.
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