Standard Form of Rational Number
A rational number can be written in many equivalent forms. For example, 2/4, 3/6, and 1/2 are all the same number. The standard form (or simplest form) is the one where the numerator and denominator have no common factor other than 1 and the denominator is positive.
Writing rational numbers in standard form makes it easier to compare them and perform operations.
What is Standard Form of Rational Number - Grade 7 Maths (Rational Numbers)?
Definition: A rational number p/q is in standard form if:
- The denominator q is positive (q > 0).
- The HCF of |p| and q is 1 (numerator and denominator have no common factor).
Steps to convert to standard form:
- If the denominator is negative, multiply both numerator and denominator by −1.
- Find the HCF of the absolute values of numerator and denominator.
- Divide both by the HCF.
Standard Form of Rational Number Formula
Standard Form: p/q where HCF(|p|, q) = 1 and q > 0
Types and Properties
Cases:
- Both positive: 6/8 → HCF(6,8) = 2 → 3/4
- Negative numerator: −9/12 → HCF(9,12) = 3 → −3/4
- Negative denominator: 5/(−10) → multiply by −1/−1 → −5/10 → HCF(5,10) = 5 → −1/2
- Both negative: −8/(−12) → multiply by −1/−1 → 8/12 → HCF(8,12) = 4 → 2/3
Solved Examples
Example 1: Positive Rational Number
Problem: Write 18/24 in standard form.
Solution:
- HCF(18, 24) = 6
- 18/24 = (18÷6)/(24÷6) = 3/4
Answer: Standard form: 3/4.
Example 2: Negative Denominator
Problem: Write 7/(−21) in standard form.
Solution:
- Make denominator positive: 7/(−21) = −7/21
- HCF(7, 21) = 7
- −7/21 = −1/3
Answer: Standard form: −1/3.
Example 3: Both Negative
Problem: Write −15/(−25) in standard form.
Solution:
- −15/(−25) = 15/25 (negative divided by negative = positive)
- HCF(15, 25) = 5
- 15/25 = 3/5
Answer: Standard form: 3/5.
Example 4: Already in Standard Form
Problem: Is 7/9 in standard form?
Solution:
- HCF(7, 9) = 1 (7 is prime, 9 = 3²)
- Denominator is positive.
Answer: Yes, 7/9 is already in standard form.
Example 5: Large Numbers
Problem: Write 36/48 in standard form.
Solution:
- HCF(36, 48) = 12
- 36/48 = 3/4
Answer: Standard form: 3/4.
Example 6: Zero Numerator
Problem: Write 0/5 in standard form.
Solution:
- 0 divided by any non-zero number is 0.
- Standard form: 0/1 or simply 0.
Answer: Standard form: 0.
Real-World Applications
Why standard form matters:
- Comparison: To compare rational numbers, convert to standard form first.
- Simplification: Answers in standard form are cleaner and easier to read.
- Exams: NCERT and CBSE expect answers in standard/simplest form.
Key Points to Remember
- Standard form requires HCF = 1 and positive denominator.
- If denominator is negative, multiply both by −1.
- Always reduce by dividing numerator and denominator by their HCF.
- 0 in standard form is 0 (or 0/1).
- Every rational number has a unique standard form.
Practice Problems
- Write 14/21 in standard form.
- Write −20/35 in standard form.
- Write 9/(−15) in standard form.
- Write −24/(−36) in standard form.
- Is 11/13 in standard form?
Frequently Asked Questions
Q1. What is the standard form of a rational number?
A rational number p/q is in standard form when the HCF of |p| and q is 1, and the denominator q is positive.
Q2. How do you convert to standard form?
Step 1: Make the denominator positive. Step 2: Find HCF of numerator and denominator. Step 3: Divide both by HCF.
Q3. Can the numerator be negative in standard form?
Yes. For example, −3/4 is in standard form. The rule is that the denominator must be positive; the numerator can be positive, negative, or zero.
Q4. Is standard form the same as simplest form?
Yes, for rational numbers they mean the same thing — the fraction reduced to lowest terms with a positive denominator.
Related Topics
- Rational Numbers
- Equivalent Rational Numbers
- Comparing Rational Numbers
- Simplest Form of a Fraction
- Rational Numbers on Number Line
- Operations on Rational Numbers
- Properties of Rational Numbers
- Rational Numbers Between Two Numbers
- Additive Inverse of Rational Numbers
- Multiplicative Inverse of Rational Numbers
- Density Property of Rational Numbers
