Comparing Real Numbers
This Grade 8 math skill from Pengi Math (Grade 8) teaches students to compare real numbers including integers, fractions, decimals, and irrational numbers using the number line and inequality symbols. Students order sets of real numbers and determine which is greater or lesser using rational approximations where needed.
Key Concepts
Property For any two real numbers $a$ and $b$, exactly one of the following is true: $a < b$ ($a$ is less than $b$), $a b$ ($a$ is greater than $b$), or $a = b$ ($a$ is equal to $b$). On a number line, the number to the right is always the greater number.
Examples To compare $ \frac{3}{4}$ and $ 0.5$, we can convert the fraction to a decimal: $ \frac{3}{4} = 0.75$. Since $ 0.75$ is to the left of $ 0.5$ on a number line, we have $ \frac{3}{4} < 0.5$. To compare $\sqrt{5}$ and $2$, we can approximate the square root: $\sqrt{5} \approx 2.236$. Since $2.236 2$, we have $\sqrt{5} 2$. To compare $ 4$ and $ \frac{17}{4}$, we can convert the fraction to a decimal: $ \frac{17}{4} = 4.25$. Since $ 4$ is to the right of $ 4.25$ on a number line, we have $ 4 \frac{17}{4}$.
Explanation Comparing real numbers means determining which number is larger, smaller, or if they are equal. A number line is a useful tool for this, as the number farther to the right is always greater. To compare numbers in different forms, like fractions and decimals, it is often easiest to convert them to the same form, usually decimals. You can then use the inequality symbols ($<, $) to state the relationship between them.
Common Questions
How do you compare two real numbers?
Place both numbers on a number line. The number further to the right is greater. You can also convert to the same form (decimal or fraction) and compare directly.
How do you compare a fraction and a decimal?
Convert the fraction to a decimal by dividing the numerator by the denominator, then compare the two decimal values.
How do you compare irrational numbers like square roots?
Use rational approximations. For example, sqrt(2) ≈ 1.414 and sqrt(3) ≈ 1.732. Since 1.414 < 1.732, sqrt(2) < sqrt(3).
What symbols are used to compare real numbers?
Use > (greater than), < (less than), and = (equal to) to compare real numbers. You can also use ≥ and ≤ for at least or at most.
Where is comparing real numbers taught in Grade 8?
Comparing real numbers is a foundational skill in the Grade 8 Pengi Math textbook.