Fractions

Definition and Basics

• A fraction is a way to show part of a whole as a ratio of two numbers
• Consists of a numerator (top number) and a denominator (bottom number)
• Numerator tells how many equal parts are being referred to
• Denominator tells how many parts the whole is divided into

Types of Fractions

• Proper fractions: numerator is less than the denominator (e.g. 1/2)
• Improper fractions: numerator is greater than or equal to the denominator (e.g. 3/2)
• Mixed numbers: a combination of a whole number and a proper fraction (e.g. 2 1/2)

Operations with Fractions

• Addition and subtraction: require common denominators
• Multiplication: multiply numerators and denominators separately
• Division: invert the second fraction (i.e. flip the numerator and denominator) and then multiply

Equivalent Fractions

• Fractions that have the same value, but different numerators and denominators
• Can be created by multiplying or dividing both numerator and denominator by the same number

Decimals

Definition and Basics

• A decimal is a way to show part of a whole as a number with a fractional part
• The fractional part is separated from the whole number part by a decimal point
• Each place value to the right of the decimal point represents a power of 10 (e.g. tenths, hundredths, etc.)

Converting Fractions to Decimals

• Divide the numerator by the denominator to convert a fraction to a decimal
• Terminating decimals: have a finite number of digits (e.g. 1/2 = 0.5)
• Repeating decimals: have an infinite number of digits that repeat in a pattern (e.g. 1/3 = 0.333...)

Operations with Decimals

• Addition and subtraction: line up the decimal points and perform the operation
• Multiplication: multiply the numbers as usual, then place the decimal point in the product
• Division: divide the numbers as usual, then place the decimal point in the quotient

Comparing and Ordering Decimals

• Compare decimals by comparing the digits from left to right
• Order decimals from least to greatest or greatest to least by comparing the digits from left to right

分数

定义和基础

• 分数是表示整体的一部分作为两个数字的比率
• 由分子（上数字）和分母（下数字）组成
• 分子表明有多少个相等的部分被参照
• 分母表明整体被分成多少部分

分数类型

• 正确分数：分子小于分母（例如 1/2）
• 不正分数：分子大于或等于分母（例如 3/2）
• 混合数：一个整数和一个正确分数的组合（例如 2 1/2）

分数运算

• 加法和减法：需要相同的分母
• 乘法：分别乘分子和分母
• 除法：将第二个分数倒过来（即交换分子和分母），然后相乘

等价分数

• 具有相同值的分数，但不同的分子和分母
• 可以通过将分子和分母同时乘或除以相同的数字来创建

小数

定义和基础

• 小数是表示整体的一部分作为一个带有小数部分的数字
• 小数部分与整数部分由小数点分隔
• 小数点右侧的每个位值表示 10 的幂（例如十分位、百分位等）

将分数转换为小数

• 除以分母将分数转换为小数
• 终止小数：具有有限位数字（例如 1/2 = 0.5）
• 循环小数：具有无限位数字的重复模式（例如 1/3 = 0.333...）

小数运算

• 加法和减法：对齐小数点然后进行运算
• 乘法：正常相乘，然后将小数点放在积中
• 除法：正常相除，然后将小数点放在商中

比较和排序小数

• 通过从左到右比较数字来比较小数
• 通过从左到右比较数字来排序小数，从小到大或从大到小