Basic Calculations: Order, Signs and Checks

Write the operation before entering the expression

Basic arithmetic uses addition, subtraction, multiplication and division to combine known quantities. For 18 items plus 6 more, use 18+6. For 24 items divided equally among four groups, use 24/4. The mathematical expression should describe the task, not merely contain its numbers. If the problem asks how many remain after giving away six, 18+6 would still calculate successfully but would answer the wrong question. Write units beside the values when they help identify the intended relationship.

Apply multiplication and division before addition and subtraction

The expression 18+6/3 gives 20 because 6/3 is evaluated first. The expression (18+6)/3 gives 8 because the parentheses group the sum. Operations of equal precedence are evaluated in their stated sequence, so 24/6*2 is (24/6)*2 = 8. If the intended denominator is the product 6*2, write 24/(6*2), which gives 2. Avoid relying on crowded slash notation when a pair of parentheses can make the grouping explicit.

Track negative values separately from the operation sign

Subtracting a negative number is equivalent to adding its positive opposite: 7-(-3) = 10. Multiplying two negative numbers gives a positive result, so (-4)*(-6) = 24. However, adding two negatives gives a negative sum: -4+(-6) = -10. The sign rules depend on the operation. Before evaluating a longer expression, determine the expected sign of each grouped product or quotient, then combine the groups according to the surrounding additions and subtractions.

Preserve a fraction with parentheses

To evaluate three fifths of 40, enter (3/5)*40, giving 24. To divide 40 by three fifths, enter 40/(3/5), giving 200/3, approximately 66.66666667 in an eight-place display. These operations are different: multiplying by a proper positive fraction reduces the amount, while dividing by that fraction increases it. For exact rational output, use the separate Fraction Calculator; a rounded decimal from the Basic Calculator should not be mistaken for the complete repeating value.

Use powers and roots with a clear base

The expression 5^2 means 5 multiplied by itself and gives 25; it does not mean 5*2. The expression sqrt(81) gives the principal square root 9. Parentheses around a negative base remove ambiguity: (-3)^2 = 9, while -(3^2) = -9. If a formula uses a squared measurement, square its unit as well. For example, a square with side 5 centimetres has area 25 square centimetres, not 25 centimetres.

Check the result through another operation

After calculating 96/8 = 12, multiply 12*8 to recover 96. After subtracting 37 from 92 to obtain 55, add 55+37 to recover 92. An estimate gives another kind of check: 49*21 should be near 50*20 = 1000, and its exact value is 1029. A result near ten or one hundred thousand would suggest a missing or extra factor. The inverse check verifies arithmetic, while the estimate can reveal a scale or transcription error.

Keep invalid operations and rounding visible

Division by zero has no finite result and should be corrected at the input or model level. A square root of a negative number is outside a real-only square-root calculation; a complex-number tool is a different workflow. Decimal arithmetic also can involve rounded displays. Keep sufficient precision through intermediate steps and round the final answer according to the task, rather than rounding every intermediate value. If the outcome represents an indivisible count, apply its stated allocation or purchasing rule after the numerical calculation.

Convert units before adding unlike scales

Adding 2 metres and 35 centimetres requires a shared unit first. Write 200+35 = 235 centimetres, or 2+0.35 = 2.35 metres. The number 37 would come from adding the displayed digits without reconciling their scales. The calculator evaluates the expression supplied, so unit consistency remains part of preparing that expression.

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