The Real Numbers All of the numbers that
The Real Numbers All of the numbers that you are currently familiar with are part of the set of real numbers.
Natural or Counting Numbers Man first used numbers to keep track of sheep, goats and other countable possessions. The Natural or Counting Numbers are the ones you use to count. { 1, 2, 3, …}
Natural or Counting Numbers There are no fractions or decimals in the counting numbers. You wouldn’t say you had three and a half sheep. You’d say you had 3 sheep and dinner!
Whole Numbers One day a very philosophical man contemplated the question: What number would I use if I had no sheep? …and that was the birth of the number zero!
Whole Numbers The set of whole numbers are the counting numbers plus their new friend, zero It is easy to remember which are the whole numbers because zero is the only number that looks like a hole! When you see the word WHOLE, { 0, 1, 2, …} think “hole”
The Integers Next, someone invented checking accounts and within the hour someone had to invent the negative numbers! The integers are the negative counting numbers and the whole numbers (still no fractions or decimals). {…-2, -1, 0, 1, 2, …}
The Rational Numbers The word ratio means fraction. Therefore rational numbers are any numbers which can be written as fractions.
Integers are Rational Numbers Like the 5 in our example, any integer can be made into a fraction by putting it over 1. Since it can be a fraction, it is a rational number.
Changing fractions to decimals It’s easy to change a fraction to a decimal, so rational numbers can also be written as decimals. Rational numbers convert to two different types of decimals: Terminating decimals – which end Repeating decimals – which repeat
Terminating decimals To convert a fraction to a decimal, divide the top by the bottom. To convert ½ to a decimal you would do: There is no remainder. The answer just ends – or terminates.
Repeating decimals To convert a fraction to a decimal, divide the top by the bottom. To convert 1/3 to a decimal you would do: = There is a remainder. The answer just keeps repeating.
Repeating decimals The bar tells us that it is a repeating decimal. The bar extends over the entire pattern that repeats.
Rational numbers as decimals Rational numbers can be converted from fractions to either • Terminating decimals or • Repeating decimals
Rational numbers The subsets of real numbers that we’ve discussed are “nested” like Russian dolls.
Venn Diagrams Rational numbers Venn diagrams illustrate how sets relate to each other. Natural numbers Whole numbers Integers Subsets are drawn inside the larger set.
Irrational Numbers In English, the word “irrational” means not rational illogical, crazy, wacky. In math, irrational numbers are not rational. They usually look wacky! …and their decimals never end or repeat!
Irrational Numbers There is one trick you need to watch out for! They look wacky but because the number in the house is a perfect square, they are really the integers 5 and 9 in disguise! Sort of like the wolf at Grandma’s house!
Rounding or truncating Some decimals are much longer than we need. There are two ways we can make them shorter. Truncating – just lop the extra digits off. Rounding – use the digit to the right of the one we want to end with to determine whether to round up or not. If that digit is 5 or higher, round up.
Truncating – just lop the extra digits off. If we want to use with just 4 decimal places. We’d just chop off the rest! 3. 1415/926… 3. 1415 Truncate ~ tree trunk ~ chop!
Rounding If we want to round to 4 decimal places. We’d look at the digit in the 5 th place 9 is “ 5 or bigger” so the digit in the 4 th spot goes up 3. 14159 3. 1416
Real numbers The set of real numbers consists of two infinite, non-overlapping sets. Rational numbers Irrational numbers Every real number is either rational or irrational, but it can’t be both!
Vocabulary Natural numbers Whole numbers Integers Rational numbers Irrational numbers Real numbers Truncating Rounding
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