Hello, and welcome to your mathematics lesson for today. We are going to explore Chapter Two: Rational Numbers. By the end of this lesson, you will understand what rational numbers are, how to write them in standard form, how to represent them on a number line, how to compare them, and how to perform all four operations—addition, subtraction, multiplication, and division—on them.
Let us begin with a simple observation. When you add two integers, you always get another integer. The same is true for subtraction and multiplication. But division is different. Take eight and five—both are integers. Eight divided by five equals 8/5, which is not an integer. Five divided by eight equals 5/8, also not an integer. These results are fractions, and they belong to a broader family called rational numbers.
Here is the precise definition.
If p and q are two co-prime integers, and q is not equal to zero, then the number p/q is called a rational number. When p and q are co-prime, they share no common factor other than one.
For example, 3/7 is a rational number because three and seven are integers, seven is not zero, and they share only one as a common factor. Similarly, -15/19 is also a rational number.
A rational number is in standard form when two conditions are met: the denominator is positive, and the numerator and denominator are co-prime, meaning they have no common factor other than one. This means the fraction must be in its lowest terms with a positive denominator. So 3/4 is in standard form. But 15/24 is not, because fifteen and twenty-four share three as a common factor.
Here are some important facts to remember. Every integer is a rational number, but not every rational number is an integer. Why? Because any integer n can be written as n/1. Five equals 5/1. Negative eight equals -8/1. Zero is also a rational number—it can be written as 0/1, or 0/5, or 0/-23.
In the fraction p/q, p is called the numerator and q is the denominator. So in 7/11, seven is the numerator and eleven is the denominator.
A rational number is positive if both numerator and denominator have the same sign—both positive or both negative. For example, 45/17 and -23/-16 are both positive rational numbers.
A rational number is negative if the numerator and denominator have opposite signs. So -5/7, 8/-11, and -23/25 are all negative. Note that -5/7, 5/-7, and -5/7 all represent the same value.
Every positive integer is a positive rational number, and every negative integer is a negative rational number. Zero is special—it is a rational number, but it is neither positive nor negative.
Now, let us learn how to represent rational numbers on a number line. First, draw a straight horizontal line and mark a point O for zero. Mark equal distances to the right for positive integers: one, two, three, and so on. Mark equal distances to the left for negative integers: negative one, negative two, negative three.
To plot a fraction like 1/2, notice the denominator is two. This tells you to divide each unit interval into two equal parts. The point halfway between zero and one represents 1/2. Similarly, 3/2 lies halfway between one and two.
For 2/3, divide each unit into three equal parts. Count two small steps to the right of zero. For -5/3, count five small steps to the left of zero.
Let me walk you through an example. Suppose you want to plot -3/2, -1/2, 1/2, 3/2, and 5/2. Since the denominator is two everywhere, divide each unit interval into two equal parts. Starting from zero and moving left, the first division point is -1/2, the second is -2/2 which equals negative one, and the third division point is -3/2 which equals negative one and a half. Moving right from zero, you hit 1/2, then 3/2 which is one and a half, then 5/2 which is two and a half.
Next, we compare rational numbers. Here are the fundamental rules. Every positive rational number is greater than zero and greater than every negative rational number. Every negative rational number is less than zero and less than every positive rational number. Zero sits in the middle—greater than all negatives, less than all positives.
To compare two positive rational numbers with different denominators, find a common denominator. Compare 3/5 and 5/7. The least common multiple of five and seven is thirty-five. Convert: 3/5 becomes 21/35, and 5/7 becomes 25/35. Since twenty-five exceeds twenty-one, we conclude 5/7 is greater than 3/5.
There is also a quick cross-multiplication method. For a/b and c/d, compare a times d with b times c. If a times d exceeds b times c, then a/b is greater. Applying this to 3/5 and 5/7: three times seven gives twenty-one, five times five gives twenty-five. Since twenty-one is less than twenty-five, 3/5 is less than 5/7.
When comparing two negative rational numbers, first compare their absolute values, then reverse the inequality. Compare -3/7 and -8/15. First compare 3/7 and 8/15. Cross-multiply: three times fifteen equals forty-five, seven times eight equals fifty-six. Since forty-five is less than fifty-six, 3/7 is less than 8/15. Therefore, -3/7 is greater than -8/15. Remember: for any two numbers, if a exceeds b, then negative a is less than negative b.
Between any two rational numbers, there exist infinitely many other rational numbers. To find rational numbers between two given values, convert them to equivalent fractions with a common denominator, then pick numerators in between.
Suppose you want numbers between -6/11 and -3/5. First, note that -6/11 is approximately negative zero point five four five, and -3/5 equals negative zero point six. Since negative zero point five four five is greater than negative zero point six, -6/11 is actually greater than -3/5. The least common multiple of eleven and five is fifty-five. Convert: -6/11 becomes -30/55, and -3/5 becomes -33/55. Now, -31/55 and -32/55 lie between them.
Now we turn to operations with rational numbers, beginning with addition.
When denominators are equal, simply add the numerators and keep the denominator. 4/15 + 8/15 equals 12/15, which simplifies to 4/5. If signs differ, follow the rules of integer addition. -4/15 + 8/15 equals 4/15.
When denominators differ, find the least common multiple and convert. Take -2/3 + 3/4. The LCM of three and four is twelve. -2/3 becomes -8/12, and 3/4 becomes 9/12. The sum is 1/12.
For multiple fractions, find the LCM of all denominators. Consider -4/9 + (-7/12) + 7/18. The LCM of nine, twelve, and eighteen is thirty-six. Convert each: -16/36, -21/36, and 14/36. Add: negative sixteen plus negative twenty-one plus fourteen equals negative twenty-three. The answer is -23/36.
Subtraction follows similar rules. With equal denominators, subtract numerators. 5/7 - 4/7 equals 1/7.
With unequal denominators, find the LCM first. Compute 3/4 - 2/5. The LCM of four and five is twenty. 3/4 becomes 15/20, 2/5 becomes 8/20. Subtract: fifteen minus eight equals seven, giving 7/20.
Subtracting a negative number is the same as adding its positive value. -2/3 - (-3/5) equals -2/3 + 3/5. The LCM of three and five is fifteen. This yields -10/15 + 9/15, which equals -1/15.
Multiplication is more straightforward.
The product of two rational numbers equals the product of their numerators divided by the product of their denominators.
Multiply 5/6 × -3/4. Numerator: five times negative three equals negative fifteen. Denominator: six times four equals twenty-four. Result: -15/24, which simplifies to -5/8.
When multiplying multiple fractions, you may simplify before multiplying. Consider (-36/7) × (28/-9). Negative thirty-six divided by negative nine equals positive four. Twenty-eight divided by seven equals four. The product is four times four, which is sixteen.
Every non-zero rational number has a multiplicative inverse, also called its reciprocal. The reciprocal of 3/4 is 4/3, because 3/4 × 4/3 equals one. The reciprocal of -4/5 is 5/-4, which equals -5/4 in standard form.
Division uses the reciprocal.
To divide by a rational number, multiply by its reciprocal. This is why division by zero is undefined: zero has no reciprocal. a/b ÷ c/d equals a/b × d/c.
Let us divide 4/25 ÷ 3/5. This becomes 4/25 × 5/3. Simplify: five and twenty-five share a factor of five. Result: 4/15.
Here is another example: 2/7 ÷ (-8/35). This becomes 2/7 × 35/-8. Thirty-five divided by seven equals five. So we have 2×5/-8, which is -10/8, simplifying to -5/4.
When both numbers are negative, their quotient is positive. (-4/3) ÷ (-16/21) becomes (-4/3) × (21/-16). Four and sixteen share a factor of four; three and twenty-one share a factor of three. This leaves 7/4.
Let me summarize the key takeaways from this lesson.
First, a rational number is any number that can be expressed as p/q where p and q are co-prime integers and q is not zero.
Second, every integer is a rational number, but not every rational number is an integer. Zero is rational but neither positive nor negative.
Third, a rational number is in standard form when its denominator is positive and the numerator and denominator share no common factor other than one.
Fourth, to compare rational numbers, convert to equivalent fractions with a common denominator, or use cross-multiplication. Remember that for negative numbers, the inequality reverses when you compare absolute values.
Fifth, between any two rational numbers, infinitely many other rational numbers exist.
Sixth, for all four operations—addition, subtraction, multiplication, and division—work with common denominators when needed, and remember that dividing by a fraction is equivalent to multiplying by its reciprocal.
You have now built a solid foundation in rational numbers. Practice converting between forms, plotting on number lines, and performing operations until these skills feel natural. Mathematics rewards patience and persistence. Keep exploring, keep questioning, and I will see you in the next lesson.