The four operations Revision Notes

Integers include positive whole numbers, zero, and negative whole numbers. When performing calculations with integers, it is essential to understand how negative numbers affect the result.

Addition and subtraction with negative numbers follow these rules: adding a negative number is the same as subtracting its positive counterpart, and subtracting a negative number is the same as adding its positive counterpart.

Multiplication and division involving negative numbers follow the sign rules: the product or quotient of two numbers with the same sign is positive; with different signs, it is negative.

• Example: Calculate (-7) + 12 = 5
• Example: Calculate 15 - (-4) = 15 + 4 = 19
• Example: Calculate (-3) × 6 = -18
• Example: Calculate (-24) ÷ (-8) = 3

Fractions represent parts of a whole and can be proper (numerator less than denominator), improper (numerator greater or equal to denominator), or mixed numbers (a whole number and a proper fraction combined).

To perform operations with fractions, it is often easier to convert mixed numbers to improper fractions first.

Addition and subtraction require a common denominator. Multiplication is straightforward by multiplying numerators and denominators. Division involves multiplying by the reciprocal of the divisor.

• Example: Add 2 1/3 + 1 3/4: Convert to improper fractions: 7/3 + 7/4. Find common denominator 12: (28/12) + (21/12) = 49/12 = 4 1/12.
• Example: Multiply 3/5 × 2 1/2: Convert 2 1/2 to 5/2. Multiply: (3/5) × (5/2) = 15/10 = 3/2 = 1 1/2.

Decimals are another way to represent fractions, especially tenths, hundredths, etc. When calculating with decimals, align decimal points for addition and subtraction.

Multiplication and division with decimals require careful placement of the decimal point in the answer.

For multiplication, multiply as if whole numbers, then place the decimal point so the total number of decimal places in the answer equals the sum of decimal places in the factors. For division, move the decimal point in the divisor to make it a whole number and move the decimal point in the dividend the same number of places, then divide as whole numbers.

• Example: Add 3.75 + 0.6 = 4.35
• Example: Multiply 2.4 × 0.3: 24 × 3 = 72, with two decimal places total, answer is 0.72
• Example: Divide 4.5 ÷ 0.15: Move decimal in divisor two places to right (15), move dividend two places (450), then 450 ÷ 15 = 30

When calculations involve more than one operation, the order in which operations are performed affects the result.

The standard order is: brackets first, then exponents, followed by multiplication and division (from left to right), and finally addition and subtraction (from left to right).

Brackets can be nested, and each set must be resolved before moving on to other operations.

• Example: Calculate 5 + 3 × (2 - 7)
• Step 1: Calculate inside brackets: 2 - 7 = -5
• Step 2: Multiply: 3 × (-5) = -15
• Step 3: Add: 5 + (-15) = -10

Temperature changes often involve positive and negative numbers and require careful use of the four operations.

For example, to find the change in temperature between two readings, subtract the earlier temperature from the later one, considering negative values if temperatures are below zero.

Multiplying or dividing temperatures is less common but may appear in contexts like averaging or scaling temperature differences.

• Example: Temperature at 8 am is -3°C, at 2 pm is 5°C. Change = 5 - (-3) = 8°C increase.
• Example: If temperature drops by 4°C every hour for 3 hours, total drop = 4 × 3 = 12°C