0580 · 1.6The four operations

Operations with negative numbers

0:00A temperature starts at negative six degrees Celsius and rises by eleven degrees. The result is positive five degrees, even though the starting value was negative. Calculations become easier when we understand what each operation does. By the end, you will understand signs, fractions, decimals and the order that keeps a calculation meaningful. The four operations are addition, subtraction, multiplication and division. Integers are whole numbers, including negative values and zero.

0:27Fractions and decimals let us describe parts of a whole. These are different ways of representing numbers, but the operations still follow the same underlying ideas. Addition combines quantities, while subtraction finds a difference or removes a quantity. Multiplication can describe equal groups or scaling a quantity. Division can mean sharing equally or finding how many groups fit. These meanings help explain the methods, rather than leaving us with rules to memorise.

0:55On a number line, adding a positive number moves you right. Adding a negative number moves you left. −2 + −3 therefore gives negative five. The negative sign belongs to the number, while the addition sign tells us which operation to perform. Subtraction reverses the movement that addition would make. Adding negative three is a jump to the left, so subtracting it is the same jump to the right. −2 − −3 therefore equals one.

1:22Subtracting a negative increases the starting value because it reverses a downward change. For multiplication and division, matching signs give a positive result, while mismatched ones make it −3 × −4 is negative twelve, since it combines three groups of negative four. A negative multiplier reverses the sign of the product. −3 × −4 is therefore positive twelve. Division follows the same sign pattern because it reverses multiplication. −12 ÷ 3 = −4, since 3 × −4 gives negative twelve.

1:58−12 ÷ −3 = 4. Division by zero is undefined, since multiplying any number by it never gives back a nonzero value.

Order of operations

2:09Calculation order makes written expressions unambiguous. Three plus four multiplied by two means three added to the product of four and two. That gives eleven, not fourteen. Brackets change the grouping, so 3 + 4 inside brackets, multiplied by two, gives fourteen. Work inside brackets first, then evaluate any powers or roots that are present. Next, perform multiplication and division from left to right. Finally, perform addition and subtraction from left to right.

2:37Multiplication does not automatically come before division, and the same holds for addition and subtraction. Our first exam-style example is negative eight plus eighteen divided by negative three, multiplied by two, minus negative five. 18 ÷ −3 gives negative six. −6 × 2 gives negative twelve. We now have −8 + −12 − −5. −8 + −12 gives negative twenty. Subtracting negative five becomes adding five. −20 + 5 gives negative fifteen.

3:12Notice that we completed division before multiplication here because those operations have equal priority and division appeared first.

Fractions and mixed numbers

3:20A fraction describes equal parts of a whole. The denominator tells us how many equal parts form the whole, and the numerator counts those selected. To add or subtract fractions, the parts must have the same size. That is why we need a common denominator. ⅓ + 1 fourth becomes four twelfths plus three twelfths. Adding the numerators gives seven twelfths because we are counting seven pieces of that size.

3:45Adding the denominators would change the size of the pieces. For subtraction, the same principle applies: rename the fractions before subtracting their numerators. An improper fraction has a numerator greater than or equal to its denominator. A mixed number combines a whole number with a proper fraction. One and ½ = 3 halves, since each whole contains two of them. Convert mixed numbers to improper fractions before multiplication or division to keep the entire quantity together.

4:12Multiplying fractions means taking a part of another quantity. ½ × ⅔ means half of that amount, which is one third. The diagram divides the whole into six equal pieces, with two in the overlap. This explains why we multiply the numerators and then the denominators. Division asks how many portions of one size fit into another quantity. ½ ÷ 1 fourth equals two, because exactly two fourths fit inside it.

4:39Dividing by a nonzero fraction is equivalent to multiplying by its reciprocal. To find the reciprocal, swap the numerator and the denominator. Our second example is one and ½ − ⅔ in brackets, multiplied by three fifths, then divided by three fourths. Convert one and one half to three halves. Rewrite three halves as nine sixths and two thirds as four sixths. Nine sixths minus four sixths gives five sixths.

5:05Five sixths multiplied by three fifths gives fifteen thirtieths. Simplify fifteen thirtieths to one half. Dividing one half by three fourths becomes multiplying one half by four thirds. This gives four sixths, which simplifies to two thirds. Only the divisor is replaced by its reciprocal when changing division to multiplication. Turning both fractions upside down would produce a different calculation. Simplifying fractions means dividing numerator and denominator by the same nonzero factor.

5:35Their values stay unchanged because the size and number of parts change proportionally.

Decimals and exam techniques

5:40For decimals, place value controls the working. Our third example is 3.6 − 1.25 in brackets, divided by zero point five, plus 0.4 × 0.3. Write three point six as three point six zero so matching place values line up. Subtraction gives two point three five. To divide two point three five by zero point five, multiply both numbers by one hundred. This gives 235 ÷ 50, which equals four point seven.

6:11Four tenths multiplied by three tenths gives twelve hundredths, or zero point one two. 4.7 + 0.12 = 4.82. Scaling both numbers in a division by the same amount leaves the quotient unchanged. Scaling only one would change the answer. For multiplication, use place value rather than lining up the decimal points. Multiplication can make a positive number smaller when the multiplier lies between zero and one. Our fourth example, a practical exam question, gives a starting temperature of negative six degrees Celsius.

6:43It rises by eleven degrees, then falls by eight. −6 + 11 gives five degrees Celsius. 5 − 8 gives a final temperature of negative three degrees Celsius. The overall change is the final temperature minus the starting one. −3 − −6 becomes −3 + 6. The result is three degrees, so the temperature rose overall. A negative final temperature does not necessarily mean the overall change was a fall.

7:13Paper two is non-calculator, so all these methods must work by hand. Paper four allows a calculator. Questions may use the command Calculate, with brackets and number types mixed together. Show intermediate working so your operation order is clear. Keep exact values during calculations and round only the final answer when required. Unless told otherwise, give answers that are not exact to three significant figures. Our examples have exact answers, so no rounding is needed.

7:42Common exam errors include ignoring brackets, adding fraction denominators and changing only one number during decimal division. Checking signs and place values helps prevent these mistakes. The main idea is to preserve the meaning of the numbers throughout the calculation. Signs describe direction, common denominators create matching fraction parts, and decimal methods depend on place value. Brackets identify grouped calculations, while equal-priority operations follow left to right. Understanding those connections makes the four operations reliable rather than mysterious.