Natural numbers are the positive counting numbers, starting with 1. {1, 2, 3, 4,...} Whole Numbers The set of whole numbers includes all the natural numbers, plus 0. {0, 1, 2, 3, 4,...} Integers The set of integers includes all the whole numbers and natural numbers and their opposites, so it includes negative numbers with no fractional part. {... −4 , −3 , −2 , −1 , 0, 1, 2, 3, 4,...} Rational Numbers The set of rational numbers includes any number that can be expressed as one integer divided by another nonzero integer, so it includes fractions and decimals. Since any integer can be expressed as a fraction with a denominator of 1, the set of rational numbers also includes all the integers, all the whole numbers, and all the natural numbers. Some examples of rational numbers are 3, 2.45, 58 , and 0. The number 3 can be expressed as 31 . The number 2.45 can be expressed as 245100 . Zero may be written as a fraction with 0 as the numerator and any number except 0 as the denominator. For example, 018 is equivalent to 0. Therefore, 0 is a rational number as well. Irrational Numbers Irrational numbers are numbers that cannot be expressed as a fraction. Some examples of irrational numbers include pi, 6–√ , and −21−−√ . Irrational numbers also cannot be expressed by an accurate and exact decimal value. When calculated as decimals, they are endless decimals with no repeating pattern.
what most important to write down
The most important information to write down from this passage is the definitions and examples of each type of number:
- Natural numbers: Positive counting numbers starting from 1.
- Whole Numbers: Natural numbers including 0.
- Integers: Whole numbers, natural numbers, and their opposites.
- Rational Numbers: Numbers that can be expressed as one integer divided by another nonzero integer, including fractions and decimals.
- Examples of rational numbers: 3, 2.45, 58, and 0.
- Irrational Numbers: Numbers that cannot be expressed as fractions and have endless decimals with no repeating pattern.
- Examples of irrational numbers: pi, square root of 6, and square root of -21.
shorter
The important information to write down includes the definitions of each type of number: natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It is also important to note examples of each type.
Rational Numbers and Irrational Numbers The sets of rational numbers and irrational numbers are mutually exclusive, and they divide the set of real numbers completely. All real numbers are either rational or irrational. There are no real numbers that are both rational and irrational. There are also no real numbers that are neither rational nor irrational. Both rational and irrational numbers are part of the set of real numbers. Therefore, all rational numbers and all irrational numbers are also real numbers.
what's important to write in this text
The important information to write down from this text is as follows:
- Rational numbers and irrational numbers are mutually exclusive and cover the entire set of real numbers.
- All real numbers are either rational or irrational.
- Real numbers cannot be both rational and irrational.
- Real numbers cannot be neither rational nor irrational.
- Both rational and irrational numbers are subsets of the set of real numbers.
- Therefore, all rational numbers and all irrational numbers are also real numbers.
shorter bullet points
- Rational and irrational numbers are mutually exclusive.
- They divide the set of real numbers completely.
- All real numbers are either rational or irrational.
- Real numbers cannot be both rational and irrational.
- Both rational and irrational numbers are subsets of real numbers.
Some decimals can be written as fractions and some cannot. That means some decimals are rational numbers, and others are irrational numbers.5 short examples
1. 0.5 can be written as the fraction 1/2, making it a rational number.
2. 0.333... (repeating decimal) can be written as the fraction 1/3, making it a rational number.
3. 0.25 can be written as the fraction 1/4, making it a rational number.
4. 0.8 can be written as the fraction 4/5, making it a rational number.
5. 0.707... (approximation of the square root of 2) cannot be expressed as a fraction, making it an irrational number.