Roman numeral notation
Ⅰ- 1 、Ⅱ-2、Ⅲ-3、Ⅳ-4、Ⅴ-5。
Ⅵ-6、Ⅶ-7、Ⅷ-8、Ⅸ-9、Ⅹ- 10
L-50，C- 100，D-500，M- 1000。
If I is placed before the letter representing a large number, it means "negative 1". IX stands for 9, which means "one less than ten".
We can still see Roman numerals at the end of some watches and TV programs (the latter indicates the production date of the program)
Roman numerals are used in Europe before the introduction of Arabic numerals (actually Indian numerals), but they are rarely used now. It came into being later than the numbers in China and Oracle Bone Inscriptions, and even later than the Egyptian decimal numbers. However, its appearance marks the progress of an ancient civilization.
binary system
Binary is a digital system widely used in computing technology. Binary data are numbers represented by 0 and 1. Its cardinal number is 2, the carry rule is "every two enters one", and the borrowing rule is "borrowing one as two", which was discovered by the German master of mathematical philosophy Leibniz in18th century.
At present, the computer system basically adopts binary, and the data is mainly stored in the computer in the form of complement. The binary in the computer is a very tiny switch, where "On" stands for 1 and "Off" stands for 0.
Decimal numbers are converted into binary numbers.
(1) The given decimal integer is divided by radix 2, and the remainder is the lowest bit of equivalent binary.
(2) Divide the quotient of the previous step by radix 2, and the remainder is the penultimate digit of the equivalent binary number.
(3) Repeat step 2 until the final quotient is equal to 0. The remainder of each division is the number of binary digits, and the last remainder is the most significant bit.
For example: (89)10 = (1011001)
Binary numbers are converted into decimal numbers:
Convert from decimal to binary and push back.
For example,10010110.
Convert from decimal to binary and push back. The highest bit is 1, which is the quotient of multiplication, division and remainder.
0x2+ 1 = 1 ......................... (the remainder is 1).
1x2+0 = 2 ...................... (remainder is 0)
2x2+0 = 4 .................. (remainder is 0)
4x2+ 1 = 9 ...................... (the remainder is 1).
9X2+0 = 18 ..................... (remainder is 0)
18x2+ 1 = 37 ...................... (the remainder is 1).
37x2+ 1 = 75 ................... (the remainder is 1).
75x2+1 =151............................. (the remainder is1).
15 1x2+0 = 302.............................................................................................................(0)
So we get the decimal number 302.
You can also transform it like this, take each one apart and multiply it by the power of 2. The last digit is multiplied by 2 to the power of 0. And so on1x28+0x27+0x26+1x25+0x24+1x23+1x222+1x2/kloc-0.
seven bridges problem
Konigsberg (present-day Kaliningrad, Russia) is the capital of East Prussia, and the famous Pledgel River runs through it.
/kloc-in the 0/8th century, seven bridges were built on this river, connecting the two islands in the middle of the river (A and B in the above picture) with the river bank. Among them, there are six islands between the island and the river bank, and the other connects the two islands.
At that time, residents had a popular pastime, that is, crossing all seven bridges at once, but no one seemed to succeed.
Then the question comes: can you cross seven bridges at the same time, and each bridge can only cross once?
Euler proved that the seven-bridge problem has no solution.
Because if the number of connected points is odd, it is called singularity, and if it is even, it is called even point. In order to draw a stroke, the middle point must be an even point, that is, the route must have another path, and the singularity can only be at both ends, so any figure can be drawn with one stroke, and the singularity either does not exist or is at both ends.
The problem of the Seven Bridges in Konigsberg is one of the famous classical mathematical problems in18th century, which is called the Seven Bridges Problem for short. It is a famous graph theory problem and an example of topology research.
Infinitely cyclic fractional component number
Infinite decimals can be divided into two categories according to whether the decimal part is cyclic or not: infinite cyclic decimals and infinite acyclic decimals. Infinitely circulating decimals cannot be divided into fractions, but infinitely circulating decimals can be divided into numbers.
So, how is the infinite cycle decimal divided into fractions? The strategy is to enlarge the infinite cycle decimal by 10, 100 or 1000 times ... so that the enlarged infinite cycle decimal is exactly the same as the original infinite cycle decimal, and then subtract the two numbers to cut off the "big tail"!
Let's look at two examples:
(1) Pure Cyclic Decimal
Divided by 0.4747 ... and 0.33 ... into components.
Think about1:0.4747 …×100 = 47.4747 …
0.4747……× 100-0.4747……=47.4747……-0.4747……
( 100- 1)×0.4747……=47
That is 99× 0.4747...= 47.
So 0.4747...= 47/99
Think 2: 0.33 …×10 = 3.33 …
0.33……× 10-0.33……=3.33…-0.33……
( 10- 1) ×0.33……=3
That is 9× 0.33...= 3.
Then 0.33...= 3/9 = 1/3.
It can be seen that the decimal part of a pure cyclic decimal can be written as follows: what is the minimum number of digits in the cyclic part of a pure cyclic decimal, and the denominator is a number consisting of several 9' s; A molecule is a number consisting of a cyclic node in a pure cyclic decimal system.
(2) Mixed cyclic decimals.
Divided by 0.4777 ... and 0.325656 ... into components.
Think about1:0.4777 ... ×10 = 4.777. ...
0.4777……× 100=47.77……②
Use ②-① to obtain:
0.4777……×90=47-4
So, 0.4777...= 43/90.
Think about 2: 0.325656×100 = 32.5656 ...
0.325656……× 10000=3256.56……②
Use ②-① to obtain:
0.325656……×9900=3256.5656……-32.5656……
0.325656……×9900=3256-32
So, 0.325656...= 3224/9900.