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数学基础与生命科学
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"Introductory Mathematics for the Life Sciences" 是一本为生命科学领域的初学者设计的数学教材,由David Phoenix撰写。这本书涵盖了基础数学概念,并将其与生物学应用相结合,旨在帮助学生建立数学技能,以理解生物科学中的计算问题。 在书中,作者首先介绍了实数和模的概念,讲解了如何进行多步运算以及乘法和加法的交换律和结合律。接下来,详细讨论了分数、百分比和比率,包括它们的运算规则和在生物学中的应用。第三章涉及基础代数和测量,涵盖了测量单位、代数表达式及其运算、方程的代入、公式转换以及不等式的基本知识。 第四章深入到幂和科学记数法,解释了如何使用幂进行乘除运算,以及如何处理分数指数。第五章介绍了浓度和准确性的概念,如体积、重量比、摩尔浓度以及显著数字的使用。第六章则讲解了数据表示,包括表格、图表、直方图和散点图等,强调了如何有效地展示和分析数据。 第七章讨论线性函数,如函数的概念、特殊线性方程和一般线性方程的求解方法,以及这些在生物光学定律(如Beer-Lambert定律)中的应用。第八章涉及幂函数,特别是二次方程的解法及其在计算pH值和反应速率中的应用。第九章介绍指数函数,包括几何序列、指数方程的求解,以及生物学中关于增长和衰减的应用。最后,第十章讲述了对数函数,定义了不同基底的对数,列出了对数的运算法则,并展示了如何用对数转换数据,特别是在处理指数和幂函数时的作用。 本书通过丰富的例题和单元测试,帮助学生巩固所学知识,并将数学概念与生物学的实际问题紧密联系在一起,是一本理想的生物学预科数学教材。
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2 INTRODUCTORY MATHEMATICS FOR THE LIFE SCIENCES
example -2 is greater than -5. Addition therefore indicates
that you move to the right, since the number is getting
bigger; subtraction indicates that you move to the left.
It is obviously important that you are able to manipulate
both positive and negative numbers. It is useful to remember
that if you are adding a negative number to a positive number
you can treat this as a subtraction, as shown in Example 1.1.
Example 1.1
It may help to remember the number line. In Example 1.1
you start at position minus two (-2) and plus three (+3) tells
you to move to the right three places, which takes you to
position one on the number line. In Example 1.2 you start at
position minus four and move one place to the left, thus
giving the answer minus five.
Example 1.2
When dealing with negative numbers, the only rule that must
be remembered is that if you subtract a negative number it
becomes positive. This can be seen in Example 1.3.
Example 1.3
A similar rule applies when multiplying or dividing; if both
numbers have the same sign the answer is positive, if their
signs are different the answer is negative. This is illustrated
in Box 1.1 and Example 1.4(a)–(c).
Example 1.4
If you subtract a
negative number it
becomes positive
Multiplying or dividing
numbers of the same
sign gives a positive
answer
![](https://csdnimg.cn/release/download_crawler_static/1354163/bg11.jpg)
NUMBERS 3
If you have more than two terms in the calculation, then to
apply the sign rules in Box 1.1 you need to break the
calculation down into parts as shown in Example 1.5.
Example 1.5
Worked examples 1.1
Evaluate:
(i) (ii) (iii) (iv)
(v) (vi) (vii) .
1.3 Modulus
On some occasions it may be the size of the value that is
important, rather than its sign. For example, suppose you
are measuring the height of a seedling in centimetres. The
exact height is 4.7 cm and you take two measurements
which are recorded in Table 1.1 along with the error.
With the first reading you have under-estimated the height
by 0.2 cm but the second reading is too large by 0.2 cm. The
error in both cases is of the same size or magnitude; it is
only the direction that is different, i.e. one is an under-
estimate and the other an over-estimate. In this case it may
be worthwhile considering the absolute values. The
absolute value takes into account the magnitude or size of
Box 1.1 Sign rules for multiplication and division.
(positive) × (positive) = positive (positive) ÷ (positive) = positive
(positive) × (negative) = negative (positive) ÷ (negative) = negative
(negative) × (positive) = negative (negative) ÷ (positive) = negative
(negative) × (negative) = positive (negative) ÷ (negative) = positive
Table 1.1
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4 INTRODUCTORY MATHEMATICS FOR THE LIFE SCIENCES
the change but it does not take into account the direction of
the change. It is denoted by two straight lines (i.e. |-2|=2)
and is usually called the modulus. In the example given
above you can say that the absolute error in both
measurements is 0.2 cm.
Worked examples 1.2
Evaluate:
(i) (ii) (iii)
(iv)
1.4 Functions with multiple operations
You often have to deal with functions which contain more
than one mathematical operation and it is important to
know in what order to perform these operations. In general,
if an expression contains brackets you always evaluate
whatever is in the brackets first, then you perform
multiplication and division and finally addition and
subtraction (Box 1.2).
Box 1.2 Priority of operations.
1 Brackets
2 Multiplication and division
3 Addition and subtraction
If there is more than one set of brackets you start on the
inside and work outwards.
Example 1.6
It is essential that these rules are applied since failure to do
so will greatly influence the outcome of the calculation, as
can be seen in the following examples.
Example 1.7
Modulus measures the
absolute value without
the sign
![](https://csdnimg.cn/release/download_crawler_static/1354163/bg13.jpg)
NUMBERS 5
Example 1.8
Note that in Example 1.8 the expressions can be rewritten to
emphasise their difference:
In general, although the list of priorities tells you which
operation to perform first, it is always best to use brackets to
clarify what is required.
Example 1.9
In Example 1.9 the brackets are not needed but their
presence can help prevent confusion and this decreases the
chance of error.
Worked examples 1.3
Evaluate:
(i) (ii) (iii)
(iv) (v)
1.5 Commutative and associative laws of addition
and multiplication
The commutative law (Box 1.3) states that:
The order in which two numbers are added or multiplied
may be interchanged.
Box 1.3 Commutative laws.
If this law holds then the order in which we add or multiply
two numbers does not matter since the order can be
interchanged. Examples 1.10 and 1.11 show this to be true.
![](https://csdnimg.cn/release/download_crawler_static/1354163/bg14.jpg)
6 INTRODUCTORY MATHEMATICS FOR THE LIFE SCIENCES
Example 1.10
Example 1.11
This law can be expanded to give the associative law. The
associative law states:
If more than two numbers are added or multiplied it does
not matter in which order they are added or multiplied.
Box 1.4 Laws of association
If, therefore, an expression contains only multiplication or
only addition, the order in which the operations are
performed is irrelevant. If you have been asked to evaluate
this type of expression you can rearrange the calculation so
that it can be performed in the easiest way possible, as
shown in Example 1.12.
Example 1.12
Both methods in Example 1.12 give the same answer but for
most people the first route would be the easier one to follow.
These rules also apply to subtraction and division, since
these are simply inverses (i.e. the opposite) of multiplication
and addition (Box 1.5).
Box 1.5 Laws of association.
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