When a decimal repeats infinitely, it is known as a repeating decimal. The repeating part of the decimal is called the repetend, while the number of times it repeats is known as the period. Repeating decimals can be either pure or mixed. Pure repeating decimals consist only of the repeating part, while mixed repeating decimals have a non-repeating part followed by the repeating part.
Understanding Decimal Numbers: Unraveling the Mysteries of Digits after the Dot
Hey there, fellow number enthusiasts! Let’s dive into the fascinating world of decimals, where numbers take on a whole new dimension after the humble decimal point.
Decimal Expansion: Digits with a Purpose
Imagine a decimal number as a line of digits, each with its own assigned place value. The digits to the left of the decimal point are the whole number part, representing units, tens, hundreds, and so on. But the fun begins after the decimal point!
Terminating Decimals: The End is in Sight
Some decimals, like 0.25 or 0.75, have a limited number of digits after the decimal point. These are called terminating decimals. It’s like hitting the finish line with no extra steps.
Repeating Decimals: The Rhythm of Numbers
But what happens when the digits just keep going and going? We’ve got repeating decimals, where a specific pattern of digits repeats endlessly. Think of it as an infinite dance of numbers.
Pure Repeating Decimals have the repeating pattern starting right after the decimal point, like 0.33333… (where the “3”s dance forever). Mixed Repeating Decimals are a bit more playful, with the pattern starting after some non-repeating digits, like 0.123456789123456789… (with the digits “123456789” repeating endlessly after the first few digits).
Period and Bar Notation: Taming the Dance
To keep track of these repeating patterns, we use overline notation, also known as the bar notation. We draw a horizontal line above the repeating digits to indicate the rhythm. For example, 0.3333… becomes 0.3. Or, for the mixed repeater 0.123456789123456789…, we’d write 0.123456789.
Irrational Numbers: The Unruly Side of Math
Ever heard of numbers that just won’t behave? Numbers that can’t be neatly written as fractions? Yes, those are irrational numbers, the rebellious outcasts of the number world!
Irrational numbers are like naughty kids that refuse to follow the rules. They can’t be represented as a simple ratio of two integers (whole numbers), which makes them a bit of a mathematical enigma. The most famous example? The never-ending, non-repeating decimal of π (pi).
Convergent Series: Bringing Order to the Chaos
But wait, there’s hope for the mathematical rebels! Convergent series are like the peacekeepers of the number realm, bringing order to the chaos of irrational numbers. These are infinite sums that magically converge to a finite limit.
One famous example is the geometric series. Picture a row of dominoes, each domino half the size of the previous one. The sum of these dominoes forms a geometric series, which converges to a finite value, even though there are an infinite number of dominoes!
Geometric Series: A Tale of Multiplication
Geometric series are like the cool kids of the series world. Each term is obtained by multiplying the previous term by a constant ratio. This multiplication party creates a series that can converge to a finite value, even if it goes on forever.
So, there you have it, the intriguing world of irrational numbers and convergent series. They may not be as easy to grasp as their rational counterparts, but they add an extra layer of complexity and excitement to the beautiful world of mathematics!
Well, there you have it folks! Now you know what it’s called when a decimal keeps going on and on, repeating itself like a broken record. It’s called a repeating decimal or a recurring decimal. So, next time you come across a never-ending decimal, you’ll know the name to call it. And remember, if you have any more questions about decimals or any other math topic, be sure to come back and visit us again. We’re always here to help!