1 Simplify each of the following: 1 简化以下每一项:
a 一个
b
c Pure 1 Section 1.5 纯 1 第 1.5 节
2 In each case, determine the number of distinct real roots of the equation . 在每种情况下,确定方程 的不同实根的数量。
a 一个
b
c Pure 1 Section 2.3 纯 1 第 2.3 节
3 For the triangle shown, find the values of: 对于所示的三角形,找出以下值:
a 一个
b
International GCSE Mathematics 国际 GCSE 数学
4 Find the solutions of , giving your answers in the form where and are integers. 找到 的解,答案以 的形式给出,其中 和 是整数。 Pure 1 Section 2.1 纯 1 第 2.1 节
5 Write in the form where and are rational numbers. 将 5 写成 的形式,其中 和 是有理数。 Pure 1 Section 1.6 纯 1 第 1.6 节
1.1 Imaginary and complex numbers 1.1 虚数和复数
The quadratic equation has solutions given by 二次方程 的解为
If the expression under the square root is negative, there are no real solutions. 如果平方根下的表达式为负,则没有实数解。
Links For the equation , 方程 的链接,
the discriminant is .
If , there are two distinct real roots. 如果 ,则有两个不同的实根。
If , there are two equal real roots. 如果 ,则有两个相等的实根。
If , there are no real roots. 如果 ,则没有实根。
You can find solutions to the equation in all cases by extending the number system to include . Since there is no real number that squares to produce -1 , the number is called an imaginary number, and is represented using the letter . Complex numbers have a real part and an imaginary part, for example . 您可以通过扩展数字系统以包括 来找到方程的所有解。由于没有实数的平方等于-1,因此数字 被称为虚数,并用字母 表示。复数具有实部和虚部,例如 。
An imaginary number is a number of the form , where . 虚数是形如 的数,其中 。
A complex number is written in the form , where . 复数以 的形式表示,其中 。
Notation The set of all complex numbers is written as . 符号 所有复数的集合写作 。
For the complex number : 对于复数 :
is the real part 是实部
is the imaginary part 是虚部
Example 1 SKILLS interPRetation 示例 1 技能 解释
Write each of the following in terms of i. 将以下每个表达式用 i 表示。
a 一个
a 一个
b
You can use the rules of surds to manipulate imaginary numbers. 你可以使用无理数的规则来操作虚数。
Watch out An alternative way of writing 注意 一种替代写法
is . Avoid writing as this can easily be confused with .
In a complex number, the real part and the imaginary part cannot be combined to form a single term. 在复数中,实部和虚部不能合并成一个单一的项。
Complex numbers can be added or subtracted by adding or subtracting their real parts and adding or subtracting their imaginary parts. 复数可以通过加或减它们的实部和加或减它们的虚部来进行加法或减法。
You can multiply a real number by a complex number by multiplying out the brackets in the usual way. 您可以通过以通常的方式展开括号,将实数与复数相乘。
Example 2 示例 2
Simplify each of the following, giving your answers in the form , where . 简化以下每一个,答案以 的形式给出,其中 。
a 一个
b
c
d
Add the real parts and add the imaginary parts. 将实部相加,将虚部相加。
Subtract the real parts and subtract the imaginary parts. 减去实部并减去虚部。
Exercise (1A SKILLS interpRetation 练习 (1A 技能 解释)
Do not use your calculator in this exercise. 在这个练习中请不要使用计算器。
1 Write each of the following in the form , where is a real number. 将以下每个写成 的形式,其中 是一个实数。
a 一个
b
c
d
e
f
g
h
i 我
j
2 Simplify, giving your answers in the form