Integers, Rational Numbers And Real Numbers
Integers: These are whole numbers, including positive numbers, negative numbers, and zero. Examples: −3, 0, 7.
Rational Numbers: These are numbers that can be expressed as fractions, where the numerator and denominator are both integers. Examples: 34, −25, 0.25 (which is 14 in fraction form).
Real Numbers: This includes all rational numbers and irrational numbers. Rational numbers can be expressed as fractions or decimals, while irrational numbers cannot be expressed as fractions and have non-repeating, non-terminating decimal expansions. Examples: π (irrational), √2 (irrational), 1.5 (rational).
In this chapter, we will be discussing the below-mentioned topics in detail
- Proper Fractions, Improper Fractions, Mixed Numbers
- Adding and subtracting fractions and mixed numbers
- Adding and subtracting negative fractions and mixed numbers
- Multiplying and dividing positive fractions and mixed numbers
- Combined operations on positive and negative fractions and mixed numbers
Rational Numbers
A rational number is a number which can be expressed as a fraction. Some examples of rational numbers are as follows:
Proper Fractions | Improper Fractions | Mixed Numbers |
---|---|---|
34 | 43 | 43=113 |
Proper Fractions:
A proper fraction is when the value of the numerator is smaller than that of the denominator.
Improper Fractions:
An improper fraction is when the value of the denominator is bigger than that of the numerator.
Mixed Numbers:
Improper fractions are expressed as the sum of an integer and a proper fraction.
Example:
113=1+13=33+13=43
Negative Rational Numbers
Proper And Improper Fractions | Mixed Numbers | |
---|---|---|
−34 | −114 | |
−34 | 3−4 | −54 |
Both −34 or 3−4 will simplify to give us −34.
This is because a negative integer divided by a positive integer would give a negative value. Likewise, a positive integer divided by a negative integer would also give a negative value.
Similarly, if we rewrite the mixed number −114 into an improper fraction, it will gives us −54 , in which it can be rewritten as −54 or 5−4.
1. Addition and subtraction of positive and negative fractions and mixed numbers
Same Denominator | Different Denominator | |
---|---|---|
Addition Step 1: Join Step 2: Expand Step 3: Simplify |
75+15=7+15=85=135 | 75+12=2(7)+5(1)10=14+510=1910=1910 |
Subtraction Step 1: Join Step 2: Expand Step 3: Simplify |
75−15=7−15=65=115 | 75−12=2(7)−5(1)10=14−510=910 |
Let’s understand this with the help of some examples:
Question 1:
Without using a calculator, evaluate the following.
- 13+(−14)
- −25−(−12)
- −312+(−13)
- 112+(−213)
Solution:
1.
13+(−14)=13−14=4(1)−3(1)12=4−312=112
2.
−25−(−12)=−25+12=−25+12=2(−2)+5(1)10=−4+510=110
3.
−312+(−13)=−72−13=−72−13=3(−7)−2(1)6=−21−26=−236=−356
4.
112+(−213)=32−73=3(3)−2(7)6=9−146=−56=−56
2. Multiplication and Division of fractions and mixed numbers
Proper & Improper fractions | Mixed Numbers | |
Multiplication | 25×34=2×35×4=620=310 | 125×34=75×34=7×35×4=2120=1120 |
Division | 25÷34=25×43=815 | 125÷34=75×43=2815=11315 |
Let’s understand this with the help of some examples:
Question 2:
Without using a calculator, evaluate the following.
- 23×(−14)
- −25×(−12)
- −312×(−13)
Solution:
1.
23×(−14)=2×(−1)3×4=−212=−16=−16
2.
−25×(−12)=−25×−12=(−2)×(−1)5×2=210=15
3.
−312×(−13)=−73×(−13)=(−7)×(−1)2×3=76=116
Continue Learning | |
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Basic Geometry | Linear Equations |
Number Patterns | Percentage |
Prime Numbers | Ratio, Rate And Speed |
Functions & Linear Graphs 1 | Integers, Rational Numbers And Real Numbers |
Basic Algebra And Algebraic Manipulation 1 | Approximation And Estimation |