Number and Operations · 3 min read
Real Numbers and the Number System · Approximating Irrational Numbers
The real number system encompasses all numbers that can be placed on a number line. Understanding the classification and relationships between different types of numbers is foundational for Grade 8 mathematics.
Rational Numbers
Rational numbers are numbers that can be expressed as a fraction p/q where p and q are integers and q ≠ 0. This includes:
To convert a terminating decimal to a fraction, place the decimal digits over the appropriate power of 10 and simplify. For example, 0.75 = 75/100 = 3/4.
To convert a repeating decimal to a fraction, use algebra. For 0.333..., let x = 0.333..., then 10x = 3.333..., and subtracting gives 9x = 3, so x = 3/9 = 1/3.
Irrational Numbers
Irrational numbers cannot be expressed as fractions of integers. Their decimal representations are non-terminating and non-repeating.
For the exam, remember that the sum of a rational and irrational number is always irrational. The product of a non-zero rational number and an irrational number is always irrational. However, the sum or product of two irrational numbers may be rational or irrational depending on the specific numbers.
Classifying Real Numbers
Every real number belongs to at least one of the categories above. The relationship is hierarchical:
Natural numbers are a subset of whole numbers
Whole numbers are a subset of integers
Integers are a subset of rational numbers
Rational numbers and irrational numbers together form the real numbers
When asked to classify a number, identify the most specific category. For instance, the number 5 is natural, whole, an integer, and rational. The most specific classification is natural number.
Since irrational numbers have infinite non-repeating decimal expansions, we often need to approximate them for practical use.
Approximating Square Roots
To approximate a square root like √50, identify the perfect squares on either side. Since 49 < 50 < 64, we know 7 < √50 < 8. Because 50 is closer to 49 than to 64, √50 is closer to 7 than to 8. A reasonable approximation is 7.07.
For more precision, use the following method. To approximate √50: 7² = 49 (too small) 7.1² = 50.41 (too big) 7.07² = 49.9849 (very close)
Therefore, √50 ≈ 7.07.
Locating Irrational Numbers on a Number Line
To place √18 on a number line, recognize that 16 < 18 < 25, so 4 < √18 < 5. Since 18 is closer to 16, √18 is closer to 4. More precisely, √18 ≈ 4.24.
For π, remember that π ≈ 3.14159. On a number line, π falls between 3 and 4, closer to 3.
Comparing and Ordering Real Numbers
When comparing rational and irrational numbers, convert to decimal approximations with sufficient precision.
To order √10, 3.2, and π from least to greatest:
√10 ≈ 3.162 3.2 = 3.200
π ≈ 3.142
The order is: √10 < π < 3.2.
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