Equal, Greater or Less Than

Equal, Greater or Less Than

As well as the familiar equals sign (=) it is also very useful to show if something is not equal to (≠) greater than (>) or less than (<)

These are the important signs to know:

=

If two values are equal, we use the "equals" sign

example: 2+2 = 4

If two values are definitely not equal, we use the "not equal to" sign

example: 2+2 ≠ 9
<

But if one value is smaller than another, we can use a "less than" sign.

example: 3 < 5
>

And if one value is bigger than another, we can use a "greater than" sign

example: 9 > 6

Less Than and Greater Than

The "less than" sign and the "greater than" sign look like a "V" on its side, don't they?

To remember which way around the "<" and ">" signs go, just remember:

  • BIG > small
  • small < BIG

The "small" end always points to the smaller number, like this:

greater than sign

Greater Than Symbol: BIG > small

 

Example:

10 > 5

"10 is greater than 5"

Or the other way around:

5 < 10

"5 is less than 10"

 

Do you see how the symbol "points at" the smaller value?

... Or Equal To ...

Sometimes you know that a value is smaller, but may also be equal to!

jug

Example, a jug can hold up to 4 cups of water.

So how much water is in it?

Until you measure it, all you can say is "less than or equal to" 4 cups.

 

To show this, we add an extra line at the bottom of the "less than" or "greater than" symbol like this:

The "less than or equal to" sign:

  

The "greater than or equal to" sign:

All The Symbols

Here is a summary of all the symbols:

Symbol
Words
Example Use
=
equals
1 + 1 = 2
not equal to
1 + 1 ≠ 1
   
>
greater than
5 > 2
<
less than
7 < 9
   
greater than or equal to
marbles ≥ 1
less than or equal to
dogs ≤ 3

 

Why would you use these?

Because there are things you do not know exactly ...

... but can still  say something about.

So we have ways of saying what you do know (which may be useful!)

 

Example: John had 10 marbles, but lost some. How many has he now?

Answer: He must have less than 10:

Marbles < 10

 

If John still has some marbles we can also say he has greater than zero marbles:

Marbles > 0

 

But if we thought John could have lost all his marbles we would say

Marbles  0

In other words, the number of marbles is greater than or equal to zero.

Combining

You can sometimes say two (or more) things on the one line:

Example: Becky starts with $10, buys something and says "I got change, too". How much did she spend?

Answer: Something greater than $0 and less than $10 (but NOT $0 or $10):

"What Becky Spends" > $0
"What Becky Spends" < $10

 

This can be written down in just one line:

$0 < "What Becky Spends" < $10

That says that $0 is less than "What Becky Spends" (in other words "What Becky Spends" is greater than "$0") and what Becky Spends is also less than $10.

Notice that ">" was flipped over to "<" when we put it before what Becky spends - always make sure the small end points to the small value.

Changing Sides

We saw in that previous example that when we change sides we flipped the symbol as well.

This: Becky Spends > $0(Becky spends greater than $0)
is the same as this: $0 < Becky Spends($0 is less than what Becky spends)

 

Just make sure the small end points to the small value!

 

Here is another example using "≥" and "≤":

Example: Becky has $10 and she is going shopping. How much will she spend(without using credit)?

Answer: Something greater than, or possibly equal to, $0 and less than, or possibly equal to, $10:

Becky Spends ≥ $0
Becky Spends ≤ $10

This can be written down in just one line:

$0 ≤ Becky Spends ≤ $10

 

A Long Example: Cutting Rope

Here is an interesting example I thought of:

Example: Sam cuts a 10m rope into two. How long is the longer piece? How long is the shorter piece?

Answer: Let us call the longer length of rope "L", and the shorter length "S"

"L" must be greater than 0m (otherwise it wouldn't be a piece of rope), and also less than 10m:

L > 0
L < 10

So:

0 < L < 10

That says that L (the Longer length of rope) is between 0 and 10 (but not 0 or 10)

 

The same thing can be said about the shorter length "S":

0 < S < 10

 

But I did say there was a "shorter" and "longer" length, so we also know:

S < L

(Do you see how neat mathematics is? Instead of saying "the shorter length is less than the longer length", we can just write "S < L")

 

We can combine all of that like this:

0 < S < L < 10

That says a lot:

0 is less that the short length, the short length is less than the long length, the long length is less than 10.

 

Reading "backwards" we can also see:

10 is greater than the long length, the long length is greater than the short length, the short length is greater than 0.

It also lets us see that "S" is less than 10 (by "jumping over" the "L"), and even that 0<10 (which we know anyway), all in one statement.

 

NOW, I have one more trick. If Sam tried really hard he might be able to cut the rope EXACTLY in half, so each half would be 5m, but we know he didn't because we said there was a "shorter" and "longer" length, so we also know:

S<5

and

L>5

We can put that into our very neat statement here:

0 < S < 5 < L < 10

And IF we thought the two lengths MIGHT be exactly 5 we could change that to

0 < S ≤ 5 ≤ L < 10

 

An Example Using Algebra

OK, this example may be complicated if you don't know Algebra, but I thought you might like to see it anyway. Just imagine that "x" is the number of people at your party, and "x+3" is what happens when 3 more people arrive.

Example: What is x+3, when you know that x is greater than 1?

If x > 1 , then x+3 > 4

 

原文:http://www.mathsisfun.com/equal-less-greater.html

转载于:https://www.cnblogs.com/xzs603/archive/2013/01/01/2841410.html

<think>好的,我现在需要解决用户提到的“索引必须大于等于零且小于参数列表大小的错误”,也就是IndexOutOfRangeException,特别是当参数列表大小为零时的情况。首先,我得理解这个错误的根本原因。 用户可能在访问一个集合或数组的元素时,使用了超出有效范围的索引。例如,如果数组长度为0,即空数组,任何索引访问都会导致这个错误。或者当参数列表为空时,试图通过索引获取参数也会触发异常。 接下来,我应该考虑常见的触发场景。比如,在C#中,访问一个空的List或数组的索引,或者在使用命令行参数时,参数列表args为空的情况下访问args[0]。或者可能是在循环中错误地计算了索引的上限,比如使用<=而不是<,导致访问最后一个元素之后的位置。 然后,我需要思考解决方案。首先,用户需要在访问索引前检查集合或数组的长度是否大于零。例如,使用if语句判断集合的Count或Length属性是否大于零。其次,要确保索引值在有效范围内,尤其是在循环中,循环条件应该严格小于集合的长度。此外,使用安全访问方法,如TryGetValue或ElementAtOrDefault,可以在不引发异常的情况下获取元素。 还需要考虑参数列表的情况。比如,在Main方法中,如果没有传入命令行参数,直接访问args[0]就会出错。因此,在访问args之前检查args.Length是否足够非常重要。 另外,异常处理也是必要的。使用try-catch块捕获IndexOutOfRangeException,可以在出现错误时进行适当的处理,而不是让程序崩溃。 可能用户的具体情况是,在某个方法中接收了一个参数列表,但在没有参数的情况下试图访问第一个元素。这时候,添加参数检查逻辑是关键。例如,在C#中: if (args.Length > 0) { // 访问args[0] } else { // 处理无参数的情况 } 或者在循环中,确保循环变量i从0到count-1,而不是等于count。 还需要提醒用户注意调试方法。比如,在出现异常时查看堆栈跟踪,确定具体哪一行代码触发了错误,检查该处的集合是否为空或索引计算是否有误。 最后,总结解决方案的步骤:检查集合是否为空,验证索引范围,使用安全访问方法,添加异常处理。通过这些步骤,可以有效避免IndexOutOfRangeException,特别是当参数列表大小为0的情况。</think>### 索引越界异常(参数列表大小为0)的解决方案 当出现$IndexOutOfRangeException$(索引越界异常)且提示“参数列表大小为0”时,通常是因为程序尝试访问空集合或数组的元素。以下是分步解决方案: --- #### 1. **检查集合/数组是否为空** 在访问索引前,先验证集合长度是否大于0: ```csharp if (args.Length > 0) // 适用于命令行参数 { string firstArg = args[0]; // 安全访问 } else { Console.WriteLine("参数列表为空"); } ``` --- #### 2. **验证索引范围** 确保索引值始终满足条件: $$0 \leq index < collection.Length$$ 例如在循环中: ```csharp for (int i = 0; i < myList.Count; i++) // 注意用<而非<= { Console.WriteLine(myList[i]); } ``` --- #### 3. **使用安全访问方法** 避免直接索引访问,改用条件判断或安全方法: ```csharp // 使用TryGetValue(适用于字典) if (myDict.TryGetValue(key, out var value)) { // 操作value } // 使用ElementAtOrDefault(LINQ扩展方法) var item = myList.ElementAtOrDefault(index); if (item != null) { // 操作item } ``` --- #### 4. **异常处理** 通过try-catch捕获异常并处理: ```csharp try { var data = args[0]; } catch (IndexOutOfRangeException ex) { Console.WriteLine($"错误:{ex.Message}"); // 恢复逻辑或日志记录 } ``` --- #### 5. **调试与日志** - **检查堆栈跟踪**:定位触发异常的具体代码行[^1] - **输出集合状态**:打印集合长度和索引值辅助调试 ```csharp Console.WriteLine($"参数数量:{args.Length}, 尝试访问索引:{attemptedIndex}"); ``` --- ### 常见触发场景 | 场景 | 错误示例 | 修复方法 | |------|----------|----------| | 空命令行参数 | `string arg = args[0];` | 添加`if (args.Length > 0)`检查 | | 空集合遍历 | `for (int i=0; i<=list.Count; i++)` | 将`<=`改为`<` | | 动态数据源 | `var item = data[data.Count - 1];` | 添加`data.Count > 0`验证 | --- ### 相关问题 1. 如何避免C#中的`NullReferenceException`? 2. 集合初始化的最佳实践有哪些? 3. 调试索引越界异常有哪些工具推荐? : 引用自编程语言异常处理机制中的基础原则 [^2]: 参考.NET框架集合操作规范 [^3]: 基于错误处理模式的多层防御策略
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