09181558

source/_posts/0004 Formal science/数学 Mathematics/math_primary.md

@@ -0,0 +1,200 @@
+---
+title: math in primary school
+categories: [0004 Formal science, 数学 Mathematics]
+tags: [math, primary school]
+---
+
+## Numbers and Place Value
+
+### What is the difference between a ‘numeral’ and a ‘number’?
+
+A numeral is the symbol, or collection of symbols, that we use to represent a number. The
+number is the concept represented by the numeral, and therefore consists of a whole network
+of connections between symbols, pictures, language and real-life situations.
+
+
+The same number (for example, the one we call ‘three hundred and sixty-six’) can be represented by different numerals – such as 366 in our Hindu-Arabic, place-value system, and CCCLXVI using
+Roman numerals
+
+Because the Hindu-Arabic system of numeration is now more or less universal, the distinction between the numeral and the number is easily lost.
+
+### What are the cardinal and ordinal aspects of number?
+
+> cardinal基数；
+> a number, such as 1, 2 and 3, used to show quantity rather than order
+>
+> ordinal序数词（如第一、第二等）
+> a number that refers to the position of sth in a series, for example ‘first’, ‘second’, etc.
+
+
+s an adjective describing a small set
+of objects: two brothers, three sweets, five fingers, three blocks, and so on. This idea of a number
+being a description of a set of things is called the `cardinal aspect of number`. 
+
+
+numbers used
+as labels to put things in order. For example, they
+turn to page 3 in a book.
+The numerals and words being used here do not represent
+cardinal numbers, because they are not referring to sets of three things.In these examples, ‘three’ is one thing, which is labelled three because of the
+position in which it lies in some ordering process. This is called the `ordinal aspect of number`. 
+
+The most important experience of the ordinal aspect of number is when
+we represent numbers as locations on a number strip  or as points on
+a number line
+
+![Alt text](/assets/images/math_primary/image.png)
+
+There is a further way in which numerals are used,
+sometimes called the `nominal aspect`. This is where
+the numeral is used as a label or a name, without any
+ordering implied. The usual example to give here
+would be a number 7 bus.
+
+### What are natural numbers and integers?
+
+use for
+counting: {1, 2, 3, 4, 5, 6, …}, going on forever.
+These are what mathematicians choose to call the set
+of `natural numbers`
+
+the set of `integers`: {…, –5, –4, –3, –2, –1, 0, 1,
+2, 3, 4, 5, …} now going on forever in both directions.
+includes both positive integers (whole numbers greater than zero) and negative integers (whole
+numbers less than zero), and zero itself.
+
+The integer –4 is properly named ‘negative four’,
+ the integer +4 is named ‘positive four’, 
+
+`natural numbers are positive integers.`
+
+![Alt text](/assets/images/math_primary/image-1.png)
+
+### What are rational and real numbers?
+
+ include fractions and decimal
+numbers (which, as we shall see, are a particular kind of(是一种特殊的) fraction), we get the set of
+`rational numbers`.
+
+The term ‘rational’ derives from the idea that a fraction represents a ratio.
+
+The technical
+definition of a rational number is any number that is the ratio of two integers.
+
+Rational numbers enable us to subdivide the
+sections of the number line between the integers and to label the points in between, 
+
+![Alt text](/assets/images/math_primary/image-2.png)
+
+---
+
+there are other real numbers that cannot be written down as exact fractions or decimals – and are therefore not rational.
+
+there is no fraction or decimal that is exactly equal to the square root of 50 (written as √50). 
+This means there is no rational number that when multiplied by itself gives exactly the answer
+50.
+
+– we could never get a number
+that gave us 50 exactly when we squared it.
+
+But √50 is a real number – in the sense that it
+represents a real point on a continuous number line, somewhere between 7 and 8. It represents
+a real length. So this is a real length, a real number, but
+it is not a rational number. It is called `an irrational number`.
+利用勾股定理得到平方根数的实际长度
+
+the
+set of real numbers includes all rational numbers – which include integers, which in turn
+include natural numbers – and all irrational numbers.
+
+### What is meant by ‘place value’?
+
+in the Hindu-Arabic system
+we do not use a symbol representing a hundred to
+construct three hundreds: we use a symbol representing three! Just this one symbol is needed to represent
+three hundreds, and we know that it represents three
+hundreds, rather than three tens or three ones, because
+of the `place` in which it is written. 
+
+in our Hindu-Arabic place-value system, all
+numbers can be represented using a finite set of digits,
+namely, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
+
+Like most numeration systems, no doubt because of the availability of
+our ten fingers for counting purposes, the system uses
+ten as a base.
+
+Larger whole numbers than 9 are constructed using powers of the base: ten, a hundred, a
+thousand, and so on.
+
+The place in which a digit is written, then, represents that number of one of these powers
+of ten
+
+for example, working from right to left, in the numeral 2345 the 5 represents
+5 ones (or units), the 4 represents 4 tens, the 3 represents 3 hundreds and the 2 represents
+2 thousands. 
+
+the
+numeral 2345 is essentially a clever piece of shorthand, condensing a complicated mathematical
+expression into four symbols, as follows:
+(2 × 103) + (3 × 102) + (4 × 101) + 5 = 2345.
+
+Perversely, we work from right to left in determining the place values, with
+increasing powers of ten as we move in this direction. But, since we read from left to right,
+the numeral is read with the largest place value first
+
+
+---
+
+the principle of `exchange`.
+This means that whenever you have accumulated ten in one place, this can be exchanged for
+one in the next place to the left. This principle of being able to ‘exchange one of these for ten
+of those’ as you move left to right along the powers of ten, or to ‘exchange ten of these for one
+of those’ as you move right to left, is a very significant feature of the place-value system.
+
+This principle of exchanging is also fundamental to the ways we do calculations with
+numbers. It is the principle of ‘carrying one’ in addition
+
+It also means that,
+when necessary, we can exchange one in any place for ten in the next place on the right, for
+example when doing subtraction by decomposition.
+
+It also means that,
+when necessary, we can exchange one in any place for ten in the next place on the right, for
+example when doing subtraction by decomposition
+
+### How does the number line support understanding of place value?
+
+![Alt text](/assets/images/math_primary/image-3.png)
+
+### What is meant by saying that zero is a place holder?
+
+‘three hundred and seven’ represented in base-ten blocks. Translated into symbols,
+without the use of a zero, this would easily be confused with thirty-seven: 37. The zero is used therefore as a **place holder**; that is, to indicate the position
+of the tens’ place, even though there are no tens
+there: 307. It is worth noting, therefore, that when we
+see a numeral such as 300, we should not think to
+ourselves that the 00 means ‘hundred’.It is the position of the 3 that indicates that it stands for ‘three hundred’; the function of the zeros is to make this
+position clear whilst indicating that there are no tens
+and no ones. 
+
+![Alt text](/assets/images/math_primary/image-4.png)
+
+### How is understanding of place value used in ordering numbers?
+
+It
+is always the first digit in a numeral that is most significant in determining the size of the number. 
+
+A statement that one number is greater than another (for example, 25 is greater than 16) or
+less than another (for example, 16 is less than 25) is called an inequality
+
+### How are numbers rounded to the nearest 10 or the nearest 100?
+
+Rounding is an important skill in handling numbers
+One skill to be learnt is to round a number or quantity to the nearest something.
+
+round a 2-digit number to the nearest ten.
+
+67 lies between 60 and 70, but is nearer to 70
+
+![Alt text](/assets/images/math_primary/image-5.png)