Arithmetic operations and operational reasoning study guide
Number • Structure and calculation
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Key concepts
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Four operations and number types
Addition and subtraction combine or remove quantities. Multiplication and division scale or partition quantities. Integers include positive and negative whole numbers. Decimals represent parts of whole units using base ten notation. Proper fractions have numerator smaller than denominator. Improper fractions have numerator greater than or equal to denominator. Mixed numbers combine a whole number and a proper fraction.
Formal written methods for whole numbers and decimals
Column addition aligns place values so digits add from right to left, causing carries when sums exceed a single place value. Column subtraction uses borrowing when a digit is too small to subtract directly. Long multiplication breaks one factor into digits and multiplies by each digit sequentially, producing partial products that sum to the total. Short (bus-stop) division and long division partition a dividend into equal groups and track remainders. The same column and long methods apply to decimals when place values align and the decimal point remains fixed.
Formal written methods for fractions and mixed numbers
Addition and subtraction of fractions require a common denominator so parts refer to the same unit; effect: denominators equalises and numerators add or subtract. Multiplication of fractions multiplies numerators and denominators directly; effect: size of parts adjusts by the product of parts. Division by a fraction uses the reciprocal of the divisor and then multiplies; effect: division becomes multiplication and simplifies calculation. Mixed numbers convert to improper fractions before multiplication or division to maintain uniform operations and avoid errors.
Negative numbers in arithmetic operations
Addition with negatives follows sign rules: like signs add and keep the sign; unlike signs subtract and keep the larger magnitude's sign. Multiplication and division use sign parity: a product or quotient is positive when factors have like signs and negative when factors have unlike signs. Formal methods proceed with absolute values while tracking sign separately so structure of column and long methods remains consistent.
Relationships between operations and inverse reasoning
Addition and subtraction are inverse operations; multiplication and division are inverse operations. Inverse relationships enable cancellation during multiplication and division of fractions and simplify algebraic expressions by removing equal factors. Inverse reasoning provides checks: performing the inverse operation returns the original value, so a subtraction followed by addition of the same amount restores the starting number.
Priority of operations and conventional notation
Conventional priority prescribes the sequence: evaluate expressions inside brackets first, then apply powers and roots, then perform multiplication and division from left to right, then perform addition and subtraction from left to right. Reciprocal notation (1/x) represents division into one unit and often appears with powers: x^(-1) equals the reciprocal. Misordering operations changes results because earlier operations alter the input for later operations.
Key notes
Important points to keep in mind