Fundamental Mathematical Operations, Signs, and Fraction Rules

Fundamental Rules of Signs and Movement on the Number Line

The interaction of mathematical signs is the first element to examine in any calculation. When two signs are placed together, such as in brackets or adjacent operations, they combine to dictate the direction of movement. Two positive signs together ($++$) result in a plus ($+$), which indicates movement to the right on a number line. Conversely, a combination of a positive and a negative sign ($+-$ or $-+$) results in a minus ($-$), indicating movement to the left. It is essential to remember that if a number has no explicit sign preceding it, that number is assumed to be positive ($+$).

Practical examples of these sign interactions include:

• $-4 + (-9) = -4 - 9$

• $-10 - (-6) = 10 + 6$

• $3 \times (-4) = -12$

• $-5 \times -2 = 10$

Hierarchical Order of Operations

The priority of mathematical operations follows a set hierarchy commonly known as BODMAS or BIDMAS. This hierarchy includes Brackets, Indices, Division, Multiplication, Addition, and Subtraction. When solving an expression, one must determine the level of operation to ensure accuracy. If all operations in a specific part of a problem are on the same level (for example, a sequence consisting entirely of multiplication and division), the calculation should be performed strictly from left to right.

Consider the multi-step example: $(6 - 16 \text{ } \div \text{ } (12 \text{ } \div \text{ } (6 - 2) + 1)) \times 3 + 1$

1. First, resolve the innermost bracket: $(6 - 2) = 4$.

2. Second, perform the division within the next bracket: $12 \text{ } \div \text{ } 4 = 3$.

3. Third, complete the addition within that bracket: $3 + 1 = 4$.

4. Fourth, divide the external number by the result of the bracket: $16 \text{ } \div \text{ } 4 = 4$.

5. Fifth, perform the subtraction within the outermost set: $6 - 4 = 2$.

6. Sixth, multiply the result: $2 \times 3 = 6$.

7. Seventh, finish with the final addition: $6 + 1 = 7$.

Fractions: Definitions, Conversions, and Core Rules

A fundamental rule of fraction manipulation is that whatever operation is performed on the denominator (the bottom number), the same operation must be performed on the numerator (the top number). Fractions are categorized by the relationship between these two numbers. A proper fraction isdefined as a fraction where the denominator is greater than the numerator. Furthermore, it is a critical mathematical rule that any number divided by zero is not zero ($n \text{ } \div \text{ } 0 \text{ } \neq 0$).

Mixed numbers can be converted into improper fractions using a specific procedure. For example, to convert the whole and proper fraction $2 \frac{1}{3}$, multiply the whole number by the denominator and add the numerator, placing the result over the original denominator: $\text{whole } 2 \text{ and } \frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{7}{3}$

Percentage and Decimal Conversions

Converting between decimals, fractions, and percentages involves moving the decimal point or performing division followed by multiplication. Multiplying a number by $100$ moves the decimal place two positions to the right. To create a percentage from a fraction, divide the numerator by the denominator and then multiply by $100$.

Examples of these conversions include:

• $0.5 = \frac{5}{10} = \frac{50}{100} = 50\%$

• $\frac{1}{4} = 0.25 = 25\%$

• $\frac{14}{25} \times 100 = 56\%$

• $0.34 \times 100 = 34\%$

• $3.3333 \dots \text{ (or } 11 \text{ } \div \text{ } 3 ) \times 100 = 366.67\%$

Mathematical Rules for Division and Sign Consistency

The Division Rule (1) states that if two numbers being divided by one another possess the same sign (both positive or both negative), the resulting answer is always positive ($+$). Conversely, the rule also dictates that if the two numbers have different signs (one positive and one negative), the resulting answer is always negative ($-$).

Indices, Powers, and Roots

Indices represent the number of times a base number is multiplied by itself. In the expression $4 \times 4 \times 4$, this is written as $4^3$, where $3$ is the power or exponent, and the result is $64$. Roots represent the inverse operation of indices. For e xample, because $3^2 = 9$, the square root of $9$ is $3$ ($\sqrt{9} = 3$).

Exponential Rules for Positive and Negative Numbers

The behavior of powers depends on whether the base is positive or negative and whether the exponent is an even or odd whole number:

1. For any positive number base, the result remains positive.

2. $(-a)^x = a^x$ if $x$ is an even whole number. For example, $(-2)^2 = -2 \times -2 = 4$.

3. $(-a)^x = -a^x$ if $x$ is an odd whole number. For example, $(-3)^3 = -3 \times -3 \times -3 = -27$.