Number and Operations · 3 min read
Understanding Rational Numbers
Rational numbers form the foundation of Grade 7 mathematics. A rational number is any number that can be expressed as a fraction p/q where p and q are integers and q is not zero. This definition encompasses integers, fractions, terminating decimals, and repeating decimals.
Integers and Their Properties
Integers include all whole numbers and their opposites: {..., -3, -2, -1, 0, 1, 2, 3, ...}. Zero is neither positive nor negative. Positive integers lie to the right of zero on the number line. Negative integers lie to the left of zero.
The opposite of a number is the same distance from zero but on the opposite side of the number line. The opposite of 5 is -5. The opposite of -8 is 8. The opposite of 0 is 0.
Absolute Value
Absolute value represents the distance a number is from zero on the number line. Distance is always positive or zero, never negative. The absolute value of 7 is 7, written as |7| = 7. The absolute value of -7 is also 7, written as |-7| = 7. The absolute value of 0 is 0.
Absolute value answers the question: How far? It ignores direction. When comparing absolute values, you compare distances. |-12| > |8| because 12 > 8. The number -12 is farther from zero than 8.
Fractions as Rational Numbers
Every fraction represents a rational number. The numerator tells how many parts you have. The denominator tells how many equal parts make one whole. Proper fractions have numerators smaller than denominators: 3/4, 2/5, 7/10. Improper fractions have numerators greater than or equal to denominators: 5/3, 8/8, 11/4. Mixed numbers combine whole numbers with proper fractions: 2 1/2, 5 3/4.
Converting between improper fractions and mixed numbers is essential. To convert 17/5 to a mixed number: divide 17 by 5 to get 3 remainder 2; the mixed number is 3 2/5. To convert 4 3/7 to an improper fraction: multiply 4 by 7 to get 28, add 3 to get 31; the improper fraction is 31/7.
Decimals as Rational Numbers
Terminating decimals end after a finite number of digits: 0.25, 3.875, 0.6. These convert directly to fractions with denominators that are powers of 10. 0.25 = 25/100 = 1/4. 3.875 = 3875/1000 = 31/8.
Repeating decimals have a digit or group of digits that repeats forever. 0.333... = 1/3. 0.666... = 2/3. 0.181818... = 2/11. A bar drawn over the repeating digits shows the pattern: a bar over the 3 in 0.3 means 0.333..., and a bar over the 18 in 0.18 means 0.181818....
To convert a repeating decimal to a fraction, use algebraic methods. For 0.333...: let x = 0.333...; then 10x = 3.333...; subtract to get 9x = 3; so x = 3/9 = 1/3.
Comparing and Ordering Rational Numbers
To compare rational numbers, convert them to the same form. Common approaches include converting all numbers to decimals or finding common denominators for fractions.
When comparing fractions with unlike denominators, find the least common denominator (LCD). To compare 3/4 and 5/6: the LCD of 4 and 6 is 12; 3/4 = 9/12; 5/6 = 10/12; since 9/12 < 10/12, we conclude 3/4 < 5/6.
On a number line, numbers increase from left to right. -5 < -2 because -5 is to the left of -2. Any positive number is greater than any negative number. Zero is greater than all negative numbers and less than all positive numbers.
To order a set of rational numbers: convert to a common form, compare systematically, arrange from least to greatest or greatest to least as specified.
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