Practice arithmetic, rational and irrational numbers, complex numbers, and quantitative reasoning with free online tests from easy to harder levels.
Arithmetic, rational numbers, irrational numbers, complex numbers, and quantitative reasoning are connected parts of one larger number system. These practice tests help you move between representations, choose valid operations, and interpret numerical results in context.
Some questions test calculation, others classification or interpretation. Recognizing which kind of numerical problem you are facing is part of the skill.
Practice accurate calculation with integers, fractions, decimals, signs, and order of operations. The focus is on preserving the structure of an expression while carrying out several operations correctly.
Work with numbers that can be written as fractions. Practice terminating and repeating decimals, equivalent representations, comparison, ordering, and calculations involving rational values.
Practice recognizing values that cannot be written as a ratio of integers. Work with non-perfect square roots, approximations, comparisons, and the distinction between exact radical form and decimal approximation.
Extend arithmetic beyond the real number line. Practice the imaginary unit i, real and imaginary parts, powers of i, and basic addition, subtraction, and multiplication of complex expressions.
Use numbers as descriptions of real quantities. Practice interpreting values in context, keeping track of units, comparing measurements, estimating reasonably, and deciding whether a numerical answer makes sense.
Quantitative reasoning asks you to treat units and scale as part of the mathematics. A result should not only be numerically correct; it should represent the right kind of quantity in the original problem.
Decide whether the number represents time, distance, area, speed, money, a ratio, or another measurable quantity.
Units can reveal whether multiplication, division, conversion, or another operation is appropriate.
An answer can be mathematically calculated but still be unrealistic because of a misplaced decimal, conversion error, or unreasonable magnitude.
State what the final number means rather than treating it as an isolated value.
Integers, finite decimals, and repeating decimals are rational. Their decimal expansions either terminate or eventually repeat.
Irrational numbers have nonterminating, nonrepeating decimal expansions. Square roots of non-perfect squares are common examples.
A calculation can look perfectly organized while still using the wrong classification, rule, or unit.
√16 = 4 is rational. Square roots of perfect squares are integers, while square roots such as √2 are irrational.
Negative signs, subtraction, and multiplication by negative values are common sources of errors in multi-step arithmetic.
Keeping units visible can prevent incorrect conversions and helps confirm whether the final quantity has the correct meaning.
Short explanations for distinctions that often cause confusion when students move between different number systems.
A rational number can be written as a fraction a/b, where a and b are integers and b is not zero. Integers, terminating decimals, and repeating decimals are rational numbers.
An irrational number cannot be expressed as a ratio of two integers. Its decimal expansion does not terminate or repeat. Common examples include π and square roots of positive integers that are not perfect squares.
No. The square root of a perfect square is rational. For example, √25 = 5. The square root of a positive integer that is not a perfect square, such as √3, is irrational.
The imaginary unit i is defined so that i² = −1. It allows equations such as x² = −1 to have solutions within the complex number system.
A complex number can be written in the form a + bi, where a and b are real numbers. Real numbers are a special case of complex numbers in which b = 0.
An expression may contain several valid operations, but performing them in a different order can produce a different value. Standard order-of-operations rules make the interpretation of an expression consistent.
Units help show what each quantity represents and can expose an incorrect setup. They are especially useful in rates, conversions, areas, volumes, and other multi-step quantitative problems.
Identify whether the error came from arithmetic, number classification, a radical or complex-number rule, comparison, units, or interpretation. Then review that specific step rather than simply repeating the entire solution.