Densityof numberline
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The document discusses the density of different number sets. It explains that the real numbers and rational numbers are dense, meaning there is always another number between any two distinct numbers. The whole numbers are not dense as there is no whole number between 1 and 2. Irrational numbers are also dense since you can always find a number halfway between any two irrational numbers by adding them and dividing by 2. Therefore, as rational numbers and irrational numbers are dense and real numbers are the intersection of these sets, real numbers must also be dense.
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1) Compare whole numbers by their place values from left to right.
2) Compare fractions with the same denominator by their numerators, and fractions with different denominators by converting them to equivalent fractions with a common denominator first.
3) Compare mixed numbers by first comparing the whole number parts, then the fractional parts if needed.
4) Compare decimals by their place values from left to right after making their decimal places the same.
5) Compare integers using the number line, with more negative numbers to the left of less negative numbers.
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The lesson plan summarizes adding rational numbers in three steps:
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Densityof numberline
- 10. Venn Diagram And the Natural Numbers are located….?
- 11. Venn Diagram
- 12. Rational Numbers
- 16. Density of the Real Numbers •The points on the real number line are dense, meaning that the real line has no gaps. •Between any two distinct points on the real number line is an infinite number of other points.
- 17. Density of the Real Numbers •The rational numbers are dense since between any two rational numbers there is always another rational number. You can always add them and divide by 2. For example 1/2 and 1/3. You can add them and divide by 2. 3/6+ 2/6=5/6 and half of that is 5/12 (5/12 is certainly between 4/12 and 6/12)
- 18. Density of the Real Numbers •The whole numbers are not dense. Is there a whole number between 1 and 2? •Irrational numbers are dense as well, as you can do the same thing you did with the rational numbers. Just add them and divide by 2, finding another number that is halfway between the two numbers.
- 19. Density of the Real Numbers Of course there are many other numbers between each rational and each irrational number. The idea of adding and dividing by two just ensures the existence of at least one such number. Now if the density property applies to rational numbers and irrational numbers, it must apply to real numbers since they can be viewed as the intersection of these two sets.
- 20. Density of the Real Numbers