NS1: Rational and Irrational numbers
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This document discusses rational and irrational numbers. Rational numbers can be written as fractions or repeating decimals, and include integers and perfect squares. Irrational numbers cannot be written as fractions and their decimals do not repeat, such as the square root of 2 or pi. Some key irrational numbers are discussed, which when calculated on a calculator either provide a non-terminating decimal or cause an error message. Rational numbers can be identified by determining if a number can be expressed as a ratio or fraction.
NS1: Rational and Irrational numbers
- 1. Unit 1 The Number System 8.NS Know that there are numbers that are not rational, and approximate them by rational numbers.
- 2. MGSE8.NS.1. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
- 3. Unit 1 The Number System MGSE8.NS.2 Use rational approximation of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line, and estimate the value of expressions (e.g., estimate π2to the nearest tenth). For example, by truncating the decimal expansion of √2 (square root of 2), show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
- 4. Part 1 Rational and Irrational numbers In the world of math, Numbers can be labeled in two different categories Rational: Any integer (positive or negative number) that can be written as a fraction/ratio and Irrational: Any integer (positive or negative number) that cannot be written as a fraction
- 7. Rational numbers Look at the word RATIONAL. What word do you see in this term? What does this world mean?
- 8. Rational numbers The term “RATIO” refers to making a comparison of two different numbers Such as 13 girls to 15 Boys or 13:15 You may remember that we can write a ratio in three ways: 13 to 15 13:15 13/15 Because we are able to write ratios in the form of a fraction. We can identify these numbers as rational.
- 9. Rational numbers What about whole numbers?? Why are they considered rational? Turn to your neighbor and discuss why 3 is considered a rational number
- 10. Rational numbers Remember that all whole numbers can be written as a fraction, just by putting a 1 underneath it. Therefore 3 can be written as a fraction and is considered rational.
- 11. Decimals and rational numbers If decimal form is another way to write a fraction, does this mean every decimal number is a fraction? This is where it can get confusing!
- 12. Decimals and rational numbers Rational numbers that are expressed as a ration that REPEAT Or TERMINATE
- 13. Terminating Decimal Think about the word terminate…What does it mean? Notice that these decimals end on their own, these numbers were not rounded. They Terminated. Terminating numbers are RATIONAL
- 14. Terminating Decimal Terminating numbers can be written as a fraction 5/10 or ½ 75/100 or ¾ 287/1000 89/100
- 15. Repeating numbers Using a calculator, you can use your phone for this!, Calculate 1 divided by 3. What did you get?
- 16. Repeating Decimals These threes will never end, they will repeat forever. To represent the repetition in a decimal, we can put a line over the first time the repeating numbers are shown and drop all the other numbers behind ig
- 17. Repeating Decimals These threes will never end, they will repeat forever. To represent the repetition in a decimal, we can put a line over the first time the repeating numbers are shown and drop all the other numbers behind it.
- 18. Repeating Decimals The process of dropping numbers without rounding is called Truncating ANY TIME YOU TRUNCATE A REPEATING DECIMAL YOU MUST PUT A LINE ABOVE ALL OF THE NUMBERS. F.Y.I. THIS LINE IS CALLED A VINCLULUM
- 19. Repeating Decimals Truncate and place the vinculum in its appropriate place.
- 20. Repeating Decimals Truncate and place the vinculum in its appropriate place. ALL REPEATING NUMBERS CAN BE TRUNCATED AND WRITTEN AS A FRACTION. THEREFORE, REPEATING NUMBERS ARE RATIONAL
- 21. Perfect Squares, numbers that are the product of a number multiplied by itself, are also rational
- 22. IRRATIONAL NUMBERS Have you ever heard someone say “you are being irrational”? What do you think that means?
- 23. IRRATIONAL NUMBERS QUICKLY TURN TO YOUR NEIGBOR AND TALK ABOUT WHY THIS IMAGE IS IRRATIONAL?
- 24. IRRATIONAL NUMBERS The term IRRATIONAL is defined as something that is illogical or unreasonable. In other other words, IT DOESN’T MAKE SENSE.
- 25. IRRATIONAL NUMBERS Let’s think about NEVER ENDS, IT GOES ON FOREVER
- 26. IRRATIONAL NUMBERS Let’s think about ALSO NEVER REPEATS.
- 27. IRRATIONAL NUMBERS IF NEVER REPEATS IF NEVER ENDS & HOW CAN WE WRITE AS A FRACTION?
- 28. IRRATIONAL NUMBERS Because we cannot write as a fraction, we call it Irrational Other numbers include the following Go ahead and put these in your calculator and see what you get.
- 29. IRRATIONAL NUMBERS You make get a long answer, or even an error message. This indicates that the number is irrational.
- 30. Rational and Irrational Differences at a Glance Rational Numbers Irrational Numbers Perfect squares Not a perfect square Fraction, unless they have a zero as a denominator Not a fraction Repeating decimals Non-Repeating decimals Terminating decimals Non-terminating decimals