Rational Numbers
Lessons in this chapter
About Rational Numbers
Rational Numbers is the direct sequel to Integer Operations: same four operations, same sign rules, same properties — just with fractions and decimals standing in for whole numbers.
Add and subtract rational numbers extends the number-line model from integers to fractions and decimals with signs, so a problem like −3/4 + 1/2 or −2.5 − 1.75 uses exactly the same combine-or-subtract logic as −4 + 7, just with a common denominator or decimal alignment step first. Use properties of addition and subtraction of rational numbers checks that commutative, associative, and identity properties still hold once the numbers are no longer whole.
Multiply and divide rational numbers reuses the same-sign-positive, different-sign-negative rule from integer multiplication, now applied to fractions (multiplying numerators and denominators, then simplifying) and decimals (multiplying as whole numbers, then placing the decimal point). Use properties of multiplication and division of rational numbers confirms the commutative, associative, and identity properties one more time, this time with fractional and decimal values.
Apply order of operations on rational numbers closes the chapter with multi-step expressions that mix all four operations on fractions, decimals, and negative numbers in a single problem — the same order-of-operations sequencing from Integer Operations, now under the added pressure of tracking denominators and decimal places alongside signs.
Frequently asked questions
Integer Operations covers whole positive and negative numbers. Rational Numbers is the same four operations and the same properties, applied to fractions and decimals with signs — a natural sequel, not new math.
A rational number is any number that can be written as a fraction p/q where q is not zero — this includes integers, fractions, and decimals that terminate or repeat, both positive and negative.
Find a common denominator first, then apply the same sign rule as integers — for −1/4 + 3/4, both fractions already share a denominator, so combine the numerators directly to get 2/4, or 1/2. Add and Subtract Rational Numbers works through cases that need a common denominator found first.
Getting the sign rule right but forgetting to simplify the fraction afterward, or forgetting that dividing by a fraction means multiplying by its reciprocal — and that the sign rule still applies to that reciprocal step.
Multiply the numbers as if they were both positive, count the total decimal places across both numbers to place the decimal point in the answer, then apply the same sign rule as integer multiplication — a negative times a positive gives a negative product.
It's not new properties — it's confirming the same commutative, associative, and identity properties from Integer Operations still hold once the numbers are fractions or decimals instead of whole numbers.