BNU1501 Basic Numeracy Flashcards

Numbers and Variables

The BNU1501 study guide from the Department of Decision Sciences at the UNIVERSITY OF SOUTH AFRICA (UNISA), Pretoria, defines numbers as abstract entities expressed through various symbols. Natural numbers, or positive integers, start at $1$ and form the set $\{1; 2; 3; 4; 5; ... \}$. Counting numbers add zero to this set, while integers include negative numbers, zero, and positive numbers. Real numbers encompass all points on a solid number line. Variables serve as placeholders for unknown numbers in algebraic expressions and equations. Common symbols include lower-case letters like $a, b, x, y$ or acronyms like $PV$ for present value.

Rounding numbers involves keeping a specified number of digits and increasing the last kept digit by one if the subsequent digit is $5$ or more. In financial contexts, rounding should only occur at the final answer to preserve accuracy, particularly with interest rates. For money, values are typically rounded to two decimal places.

Basic Operations and Exponents

Basic arithmetic includes addition, subtraction, multiplication, and division. Addition and multiplication are unique and commutative, meaning the order of numbers does not change the result: $6 + 2 = 2 + 6$ and $6 \times 2 = 2 \times 6$. Subtraction and division are not unique, requiring strict attention to order. The priority rules (BODMAS) dictate the order of operations: Brackets, Powers and Roots, Multiplication and Division (left to right), and finally Addition and Subtraction (left to right).

Exponentiation is repeated multiplication, where $a^n$ represents a base $a$ multiplied by itself $n$ times. Key laws include $a^m \times a^n = a^{m+n}$, $\frac{a^m}{a^n} = a^{m-n}$, and $(a^m)^n = a^{m \times n}$. Any non-zero base to the power of zero equals one ($a^0 = 1$), and negative exponents indicate an inverse: $a^{-n} = \frac{1}{a^n}$. Roots perform the inverse operation of exponents; for variables, the square root of $x^8$ is $x^4$ because $x^4 \times x^4 = x^8$.

Factors, Fractions, and Ratios

A factor is an integer that divides another number without leaving a remainder. Prime numbers, such as $2, 3, 5, 7$, have only two factors. Factors are used to determine the Lowest Common Multiple (LCM), which is the smallest number that is a multiple of two or more given numbers. Ratios compare quantities, such as a pupil-to-teacher ratio of $40:1$, and are calculated by dividing a total into equal parts defined by the sum of the ratio components.

Fractions consist of a numerator and a denominator. Operations on fractions require specific rules: multiplication involves multiplying numerators and denominators separately, while division requires inverting the divisor and multiplying. Addition and subtraction require a common denominator, typically the LCM. Decimals are base-$10$ representations of fractions, where percentages indicate values per hundred ($x\% = \frac{x}{100}$).

The International System of Units (SI) and Measurement

Measurement in South Africa follows the Système international d’unités (SI). The base unit for length is the metre ($m$). Standard conversions include $10\,mm = 1\,cm$, $100\,cm = 1\,m$, and $1\,000\,m = 1\,km$. Perimeter ($C$) measures the distance around a boundary, such as $2(l + w)$ for rectangles or $2\pi r$ for the circumference of a circle. Area ($A$) measures surface size in square units ($m^2$), with formulas including $l^2$ for squares, $l \times w$ for rectangles, $\frac{1}{2}bh$ for triangles, and $\pi r^2$ for circles. Land is often measured in hectares ($ha$), where $1\,ha = 10\,000\,m^2$.

Volume ($V$) measures capacity in cubic units ($m^3$). Formulas include $l^3$ for cubes, $l \times b \times h$ for prisms, and $\pi r^2 h$ for cylinders. Volume is often expressed in litres ($\ell$), where $1\,m^3 = 1\,000\,\ell$ and $1\,\ell = 1\,000\,cm^3$.

Equations and Straight Lines

Solving an equation involves manipulating it to isolate an unknown variable on the left-hand side using inverse operations. Changing the subject of a formula allows for re-arranging relationship recipes to solve for different independent variables. A coordinate system uses a horizontal x-axis and a vertical y-axis intersecting at the origin $(0; 0)$. Straight lines are represented by the functional equation $y = bx + a$, where $b$ is the slope (gradient) and $a$ is the y-intercept. The slope is the ratio of the change in y-values to the change in x-values: $b = \frac{y_1 - y_2}{x_1 - x_2}$.

Financial Interest and Amortisation

Simple interest is calculated using $I = Prt$, and the future value is $S = P(1 + rt)$. Compound interest, where interest is earned on both the principal and previously earned interest, uses $S = P(1 + i)^n$. In this formula, $i$ is the interest rate per compounding period and $n$ is the total number of periods. Annuities involve equal payments ($R$) at equal intervals. The future value of an annuity is $S = R\left[\frac{(1 + i)^n - 1}{i}\right]$, and the present value is $P = R\left[\frac{(1 + i)^n - 1}{i(1 + i)^n}\right]$.

Mortgage loans are typically amortised, meaning the principal and interest are paid off in equal installments. An amortisation schedule tracks the outstanding balance, the interest paid on that balance, and the portion of the payment that reduces the principal. Tools such as the SHARP EL-738FB calculator are recommended for performing these complex financial calculations precisely.