Mathematics in middle school is inseparable from calculation. If you develop some good or fast calculation habits in the process of learning, you will not only bring convenience to yourself in mathematical calculation, but also have a lot of convenience in life. Here are some methods for Nanshan students' reference.

Method 1: Remember the common squares and cubes:

For example, 12 = 1, 22 = 4, 32 = 9, ...,102 =100, ..., 272 = 729, ... as far as possible! (See Method 4) 13 = 1, 23 = 8, 33 = 27, 43 = 64, 53 = 125, 63 = 2 16, 73 = 343, ... Please think about it, do we live in a three-dimensional space? Cubes are everywhere.

Method 2: Speed-shifting algorithm: The result can often be obtained quickly in the calculation by appropriately shifting the factor or decimal point of a number or part of a number.

Example 1, simple shift speed algorithm; For example, 32× 125 = 4000. The algorithm is to remove the factor 8 from 32 and multiply it by 125 to get 1000. The answer is 4000 immediately! Another example is 48 ×25 = 1200. The algorithm is to remove the factor 4 from 48 and multiply it by 25 to get 100. We will soon know that the answer is 1200!

For example, 1. 84 × 25 = ___________.2.64 × 125 = ___________.3. 120 ×25 = _________.

4. 124 × 25 = __________.5.24 × 125 = ____________.6.440 × 125 = _________.

Note: 1. Generally speaking, there is a factor of 4 in the multiplicand. 25 shifts by 4, and he is100,250 shifts by 4, and he is 1000.

2. There is a factor 8 in the multiplicand. If you meet 1.25, you get him10; If you meet 12.5, you will get him100; If you meet 125, you will get him 1000.

Example 2, Example 1 What if there are no 4 and 8 factors in the multiplicand? Let's multiply 100 by 4 and 1000 by 8.

For example: 92×25 = 9200 ÷ 4 = 2300.

802 × 125 = 802000 ÷ 8 = 100250

38 × 25 = 3800 ÷ 4 =950

46 × 125 = 46000 ÷ 8 = 5750

For example, 1. 82 × 25 = ___________.2.68 × 125 = ___________.3. 122 ×25 = _________.

4. 126 × 25 = __________.5.44 × 125 = ____________.6.444 × 125 = _________.

7. 18 × 35 = _________ .(= 9×70=630) 8. 14 × 75 = _______.9. 12 × 45 =_______.

Example 3: Another example is 998+474 = 1472. The algorithm is to remove 2 to 998 and simply get 1472,,,

How many shift speed algorithms are waiting for you to find, and your computing power has been improving!

Example 4: Calculate 7.53× 0.1+75.3× 0.5+753× 0.049 = 753× (0.001+0.05+0.049) = 753× 0.1= 75.

Method 3: Pay attention to the application of fractional and decimal exchange:

For example, 32×75 = 32×2400.

For example: 68× 750 = 68×××1000 = (68÷ 4 )× 3×1000 =17× 3×100 = 5100.

For example, 84× 0.75 = 84× = (84÷ 4 )× 3 = 2/kloc-0 /× 3 = 63.

Note: 1. Generally speaking, there is a factor of 4 in the multiplicand. In the case of 75, the multiplicand is divided by 4, multiplied by 3, and then added with two zeros. In the case of 750, the multiplicand is divided by 4, multiplied by 3, and then added with three zeros. In the case of 7.5, the multiplicand is divided by 4, then multiplied by 3, plus a zero.

2, can make good use of,,,, 0.875 =

For example, 480×125 = 60×1000 = 60000, 24×375 = 24000×3000×3 = 9000, and 8×625 = 8000× 1000×5.

Exodus 64× 625 = _ _ _ _ _ _. 96× 62.5 = _ _ _ _ _ _. 32× 0.625 = _ _ _ _ _ _ _.

Method four, the application of simple formula:

For example, 1, 98×102 = (100–2 )× (100+2) =10000–4 = 9996. (Application (a+b)(a-b)=a2-b2)

Example 2, if the type is (10x+5)2, you can get (x+ 1)(x)25, for example, 752 =(7×8) and then write 25 = 5625,452 = 2025 ... The reason is (1).

Example 3, use the formula (10a+b) 2 = a2×100+B2+2a× b×10.

( 17)2 = 149+ 140 = 289

( 18)2 = 164 + 160 = 324

(27)2 = 22× 100+72 + 2×2×7× 10= 449+280=449+300-20=729

(39)2 = 32× 100 + 92 + 2×3×9× 10 = 98 1 + 540 = 152 1

Example: mental arithmetic 192, 232, 242, 262, 282, 292,,,

Example 4. The square number can also be calculated by the following formula: A2 = (a+b) (a–b)-B2.

For example: 392 = (39+1) (39-1)+1= 38× 40+1=1521.

262 =(26+4)(26-4)+ 16 = 22×30+ 16=676

272 = 24×30+9= 729

Example 5. The product of two not too big consecutive numbers: n× (n+1) = N2+n.

For example: 26×27 = 676+26 = 702,12×13 =144+12 =156,

Example 6: The square root of four consecutive integers multiplied by 1 is equal to the product of the middle two numbers minus 1.

=

For example, the value of. It is 2002× 2003–1= 4010005.

Example 7. When the ten digits of two digits are opposite to the two digits of one digit, it is only necessary to calculate the result of subtracting the ten digits ×9.

For example, 73–37 = 4× 9 = 36, 84–48 = 4× 9 = 36, 93–39 = 6× 9 = 54,,

The reason is (10× a+b)–(10× b+a) =10 (a-b)–(b-a) = (a-b )× 9.

Similarly; When two three-digit opposites are subtracted, only the subtraction result of 100 digits ×99 is needed.

Such as 783–387 = 4× 99 = 396, 947–749 = 2× 99 =198, 835–538 = 297,, (for reference, 396+963 =1089,65438.

Example 8. When ten digits of a two-digit number and two digits of a one-digit number are added in opposite directions, only the addition result of ten digits is required × 1 1.

Such as 34+43 = 7× 1 1 = 77, 49+94 =13×1=141,78+87.

Note: Multiplying a number by 1 1 only needs to add the middle two digits and the two digits on both sides. For example,14×1=151,12×1=132,/kloc.

For example, the observed value is 9×8=72.

99×98=9702

999×998=997002

9999×9998=99970002

…………………………………………………………………………… ..

Trial calculation:1.9999999999× 999999998 = _ _ _ _ _ _. A: 9999997000002

2.9999999999× 99999997 = _ _ _ _ _ _ _ _ _ _ _. A: 99999960000003

3.999999× 999994 = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _. A: 999300006

4.9999× 9992 = _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _. Answer: 999 10008.

For example,1+2+3+4+5+6+7+8+7+6+5+4+3+2+1= 8× 8 = 64. Think of it as an 8×8 square area.

For example, the calculation method of1+3+5+…+(2n-1) = N2 is the same as the above example.

Method 5: Calculate the sum of continuous arithmetic numbers. Intermediate number × number

Example 1, 1+3+5+7+9 = 5×5 = 25 (odd hours)

Example 2, 3+5+7+9+1+13 = 8× 6 = 48 (even hours)

Method 6: Add and subtract reference numbers.

3 1 + 32 + 29 + 30 + 27 + 33 + 28 = 7×30 + ( 1 + 2 – 1 + 0 – 3 + 3 – 2) = 2 10

This method is often used to calculate statistics, also known as translation method.

Method 7: Use of complement (formula).

Example 1, 9+99+999+99999+99999 =100+1000+10000+10000+/klc.

Example 2, 22+23+24+…+210 =1+2+22+23+24+…+29 = 210–2–1=/kloc-0.

Example 3: ω is the complex root of x10–1= 0. What is the value of ω +ω2+ω3+ω4+…+ ω9?

Since 1+ω+ω2+ω3+ω4+…+ ω9 = 0, ∴ω +ω2+ω3+ω4+…+ ω9 =- 1.

Note: The above methods are used in many places!

Example 4, (2+1) (22+1) (24+1) ... (2n+1) =?

Complete a bracket (2–1) (2+1) (22+1) (24+1) … (2n+1) = 22n–1.

Another example is evaluation.

Add a bracket = 1-.

Method eight. Application of some key figures:

For example, you know 7 ×1×13 =1001.

Then 479× 7×1/kloc-0 /×13 = 479479.

Others are1/kloc-0 /×1=11,1×165438. 1 1× 1 1× 1 1= 1 1× 12 1= 133 1,

1 1× 13 1= 144 1, 1 1× 14 1=3×5 17=3× 1 1×47= 155 1, 1 1× 15 1= 166 1, 1 1× 16 1= 177 1, 1 1× 17 1 = 1 1× 3× 19 = 188 1, 1 1.

Method 9. Appropriate use of exchange method, combination method and distribution method to calculate quickly: (in fact, it is the same as moving position method)

For example, 8000 ÷125 ÷ 8 = 8000 ÷ (125× 8) = 8-using the associative law.

For example, 8000000 ÷125 ÷ 5 ÷ 25 ÷ 8 ÷ 4 ÷ 2 = 800000000 ÷ [(125× 8) (25× 4) (5

For example, 256 ÷ 72×18 ÷ 4 = 256 ÷ (72 ÷18× 4) = 256 ÷ (4× 4) = 256 ÷16. Note that when parentheses are placed before the division symbol, the multiplication and division symbols in parentheses should be interchanged.

For example, 4500 ÷ 25 = 45×100 ÷ 25 = 45× (100 ÷ 25) = 45× 4 =180.

For example, 45000 ÷125 = 45×1000 ÷125 = 45× (1000 ÷125) = 45× 8 =

For example, 999+999× 999 = 999× (1+999) = 999000-using the distribution law.

For example, 9999× 9999+19999 = 9999× 9999+(10000+9999) =10000+9999× (9999+1) = 60.

Others:

Understand the nature of 5, 15, 25, 35, 45, 55, 65, 75, 85, 95;

1, a number multiplied by 5, the calculation method is to multiply 10 first, and then divide by 2, which is faster.

For example, 7348×5=73480÷2=36740. Because it is easier to divide a number by 2 than multiply it by 5, what do you think?

2. Multiply a number by 15. The calculation method is to add up half of the numbers first, and then it becomes 10, which is faster.

For example, 2242×15 = (2242+11)×10 = 33630. Because 2242× 15 = 2242×1.5×10, multiplying by15 is to add half the original number.

3. It is faster to divide a number by 4, then multiply it by 100, and then multiply it by 25.

For example, 2484× 25 = (2484 ÷ 4 )×100 = 62100. Because 2484× 25 = (2484×100) ÷ 4 = (2484 ÷ 4) ×100.

Multiply a number by 35, 45 and 55. The calculation method is to multiply the number by 2 and divide it by 2 faster.

For example,123× 45 =123× 90÷ 2 =11070÷ 2 = 5535.

5. Multiply a number by 75. It is faster to divide this number by 4 and then multiply it by 300.

For example, 284× 75 = 7/kloc-0 /× 3×100 = 21300.

6. As for a number multiplied by 55, 65, 75, 85, 95, some convenient algorithms can also be found.

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Do you have any experience in fast calculation yourself? Add some of your personal calculations to him!

Pay attention to common sense: China takes (100), ten (10 1), hundred (102), thousand (103), ten thousand (104), and so on. (1) (10/28), Tu (10256), Song (105 12), Jian (10 1024). Do you know that?/You know what?