What Are Integers?
Integers are whole numbers that contain no fractional or decimal components. They span the entire number line: positive integers like 5 and 2,847, negative integers like −3 and −156, and zero itself.
The defining characteristic is completeness without parts. You can write 8 as a fraction (8/1), but 3.5 or π are not integers because they require fractional notation to express accurately.
Common examples include:
- Positive integers: 1, 42, 9,001, 1,000,000
- Negative integers: −7, −99, −15,342
- Zero: 0 (neither positive nor negative)
Every integer qualifies as a rational number because you can express it as a fraction with an integer numerator and denominator (e.g., 5 = 5/1). However, not all rational numbers are integers—fractions like 1/2 sit between integers on the number line.
Understanding the Number Line and Sign Rules
The number line visualizes integers as points extending infinitely in both directions from zero. Positive integers increase to the right; negative integers decrease to the left. This spatial arrangement reveals why sign rules matter.
For addition and subtraction: Start at integer a, then move b steps right (if b is positive) or left (if b is negative). Subtraction reverses the direction: move left for positive b, right for negative b.
Examples:
- 7 + (−3) = 4: Start at 7, move 3 left
- −2 − 4 = −6: Start at −2, move 4 left
- −5 − (−8) = 3: Start at −5, move 8 right
For multiplication and division: The sign of the result depends on the signs of the factors (or dividend and divisor). If both numbers share the same sign, the result is positive. If they have opposite signs, the result is negative. The magnitude is calculated as if both were positive.
Core Integer Operations
The calculator evaluates seven fundamental operations. Input integers a and b to compute:
Addition: a + b = res1
Subtraction: a − b = res2
Multiplication: a × b = res3
Division: a ÷ b = res4
Exponent: a^b = res5
Root: ᵇ√a = res6
Logarithm: log_b(a) = res7
a— The first integer (base in exponentiation and logarithm, dividend in division)b— The second integer (added/subtracted, exponent, root index, or logarithm base)res1–res7— Results of each operation in sequence
Exponents, Roots, and Advanced Operations
Exponents repeat multiplication. A positive exponent means multiply the base by itself that many times: 3² = 3 × 3 = 9. For negative bases with even exponents (e.g., −2⁴), the result is positive because pairs of negatives cancel. Odd exponents (e.g., −2³ = −8) preserve the negative sign.
Roots reverse exponentiation. The square root (²√) asks "which number multiplied by itself gives this?" Crucially, even roots of negative integers are undefined in real numbers—there's no real number whose square is negative. Odd roots of negatives work fine: ³√(−8) = −2.
Logarithms ask "what power must we raise the base to, to get this number?" For example, log₂(8) = 3 because 2³ = 8. Logarithms of zero or negative numbers are undefined in standard mathematics.
Common Pitfalls and Practical Tips
Avoid these frequent mistakes when working with integers:
- Watch the sign in subtraction and negative numbers — Subtracting a negative is addition: 5 − (−3) = 5 + 3 = 8. Students often flip signs incorrectly when double negatives appear. The number line method eliminates confusion: start at 5, move 3 positions right.
- Remember even exponents always yield positive results — (−4)² = 16, not −16. The negative sign is part of the base, not separate. Conversely, −4² = −16 because the exponent applies only to the 4. Parentheses matter.
- Even roots of negative numbers don't exist in real numbers — Square roots, fourth roots, and all even-indexed roots of negatives have no real solution. Odd roots work fine: ³√(−27) = −3. Don't confuse root index with the base's sign.
- Division by zero is undefined — No integer divided by zero produces a valid result. If your calculator shows an error, check that the divisor or logarithm base isn't zero or one (log_b(a) requires b ≠ 1 as well).