Comparing Real Numbers via Decimals
Comparing real numbers via decimals is a Grade 11 Algebra 1 skill from enVision Chapter 1 that converts all real numbers to decimal form before comparing. Convert fractions (3/4 = 0.75), irrationals (sqrt(2) = 1.414), and decimals, then compare digit by digit left to right. To compare -5/3, -1.8, and -sqrt(3): their decimals are -1.667, -1.8, and -1.732, giving the order -1.8 < -sqrt(3) < -5/3. Comparing 22/7 vs pi: 22/7 = 3.14286 and pi = 3.14159, so pi < 22/7 — demonstrating that common approximations are not exact.
Key Concepts
To compare and order real numbers, convert each number to its decimal form and compare digit by digit from left to right. For any two real numbers $a$ and $b$: $a < b$ if the decimal representation of $a$ is less than the decimal representation of $b$.
Common Questions
How do you compare a fraction and an irrational number?
Convert both to decimal form. For example, 7/3 = 2.333 and sqrt(5) = 2.236, so sqrt(5) < 7/3.
Order -5/3, -1.8, and -sqrt(3) from least to greatest.
Decimals: -5/3 = -1.667, -1.8 = -1.8, -sqrt(3) = -1.732. Order: -1.8 < -sqrt(3) < -5/3.
Is 22/7 equal to pi?
No. 22/7 = 3.142857... and pi = 3.14159..., so pi < 22/7. The difference is about 0.00126.
How do you compare decimals digit by digit?
Start with the whole number part, then tenths, hundredths, and so on. The first position where digits differ determines the order.
Why are negative irrational numbers harder to order?
For negatives, larger absolute value means smaller number. You must convert and compare decimal magnitudes, then apply the reversal for negatives.
To how many decimal places should you approximate irrationals for comparison?
Enough to distinguish them. If numbers agree in tenths and hundredths, carry to thousandths. Always use more places than the difference requires.