Rational numbers are numbers that can be written as a fraction, where both the top number (numerator) and the bottom number (denominator) are whole numbers. This means that all whole numbers are also rational numbers. For example, the number 1 can be shown in different ways, like:
These examples show that there are many ways to write the same rational number.
Let’s look at the number -7. It can be written as:
These examples prove that -7 is a rational number.
Rational numbers aren’t just whole numbers. For example, the decimal 3.75 can also be written as a fraction. Here are some ways to do it:
These examples show that 3.75 is a rational number.
Repeating decimals are also rational numbers. A common example is 0.333…, which can be written as ( 0.overline{3} ). This is the same as ( frac{1}{3} ). Another example is 0.666…, which equals ( frac{2}{3} ).
In fact, any repeating decimal can be turned into a fraction.
Not all numbers are rational. Some numbers, called irrational numbers, cannot be written as a fraction of two whole numbers.
Some famous irrational numbers include:
Even though irrational numbers might seem rare, they are actually quite common. There is always at least one irrational number between any two rational numbers. This means there are just as many irrational numbers as there are rational ones.
In summary, rational numbers include all whole numbers, decimals that end, and repeating decimals. On the other hand, irrational numbers, like π and e, cannot be written as fractions. The fact that there are irrational numbers between any two rational numbers shows how common they are, making them an important part of math to understand.
Think of a whole number and write it as a fraction in three different ways. For example, if you choose the number 5, you could write it as ( frac{5}{1} ), ( frac{10}{2} ), and ( frac{15}{3} ). Share your examples with a partner and discuss how each fraction represents the same rational number.
Choose a decimal number that ends or repeats, such as 0.75 or 0.666…, and convert it into a fraction. Show your work and explain your process to the class. This will help you understand how decimals can be expressed as rational numbers.
Research one famous irrational number, such as π or the square root of 2. Create a poster that includes interesting facts about the number, its history, and where it is used in real life. Present your poster to the class to share your findings.
Go on a number hunt around your home or school. Find examples of rational and irrational numbers in everyday life, such as measurements, prices, or objects. Write down your findings and classify each number as rational or irrational. Share your list with the class and discuss any surprising discoveries.
Draw a number line and mark several rational numbers on it. Then, identify and mark at least two irrational numbers between each pair of rational numbers. This activity will help you visualize the abundance of irrational numbers and their relationship to rational numbers.
Rational – A number that can be expressed as a fraction where both the numerator and the denominator are integers, and the denominator is not zero. – Example sentence: The number 3/4 is a rational number because it can be expressed as a fraction.
Irrational – A number that cannot be expressed as a simple fraction, meaning its decimal form is non-repeating and non-terminating. – Example sentence: The number π (pi) is an irrational number because its decimal form goes on forever without repeating.
Numbers – Symbols or words used to represent quantities and used in counting and calculations. – Example sentence: In algebra, we often use letters to represent numbers in equations.
Fraction – A mathematical expression representing the division of one integer by another. – Example sentence: The fraction 1/2 represents one part of a whole that is divided into two equal parts.
Decimal – A number expressed in the base-10 numeral system, which uses a decimal point to separate the whole number from the fractional part. – Example sentence: The decimal 0.75 is equivalent to the fraction 3/4.
Repeating – A decimal in which a digit or group of digits repeats infinitely. – Example sentence: The decimal 0.333… is a repeating decimal because the digit 3 repeats indefinitely.
Square – The result of multiplying a number by itself. – Example sentence: The square of 5 is 25 because 5 multiplied by 5 equals 25.
Root – A value that, when multiplied by itself a certain number of times, gives the original number. – Example sentence: The square root of 16 is 4 because 4 times 4 equals 16.
Examples – Specific instances that illustrate or explain a general mathematical concept or rule. – Example sentence: Examples of rational numbers include 1/2, 3, and -4.5.
Properties – Characteristics or rules that apply to numbers or operations in mathematics. – Example sentence: The commutative property of addition states that changing the order of numbers does not change their sum.
