Read on to learn all about basic mathematical calculations, including division, exponents, and algebraic equations.
What Is the Real Number System?
The real number system has natural numbers, whole numbers and integers.
Natural numbers are the set of counting numbers. For example, numbers like 1, 2, 3, 4, 5, 6, etc. are natural numbers.
Whole numbers are the set of numbers that include zero (0) plus the set of natural numbers. For example, {0, 1, 2, 3, 4, 5} is a set of whole numbers.
Integers are a set of whole numbers and their opposites. For example, {-5, -4, 0, 3, 6} is a set of integers. Integers are used to count whole objects rather than fractions or parts of them. An integer that is greater than zero (e.g., 1, 2, and 3) is called a positive integer, while an integer that is less than zero (e.g., -7, -9, and -10) is called a negative integer.
Real numbers consist of all rational and irrational numbers.
Rational numbers are any numbers that can be expressed in the form of a/b, where a and b are integers and b ≠ 0. For example, numbers like 1/4 and 2/3 are examples of rational numbers.
On the other hand, Irrational numbers are any numbers that cannot be expressed as a/b. For instance, √2 and √7 are irrational numbers.
Rounding Off of Numbers
Some arithmetic calculations often result in answers given in decimals. Examples include 6.15612, 2.3464, and so on. When your results are expressed in decimals and not in whole numbers, you will have to round them off. Rounding off numbers is often seen as confusing and arbitrary. However, the following rules of thumb can help you in the rounding off of numbers:
- When a decimal is followed by numbers greater than 5, round up (e.g., 3.26 is 3.3) and when it is followed by a number less than 5, round down (e.g., 3.21 is 3.2).
- When an odd decimal is followed by a 5, round up (e.g., 2.75 is 2.8) and when an even decimal is followed by a 5, round down (2.65 is 2.6).
- Decimals generally are rounded off to the 10th place (e.g., 2.56 is 2.5), 100th place (e.g., 2.563 is 2.56), or 1000th place (e.g., 2.5638 is 2.564).
As with all rules, the rules of rounding need to be applied with good judgment. In situations requiring great precision, rounding to the 1000th or even to the 10000th may be required.
Mathematical Operations Performed on Numbers
Four types of mathematical operations can be performed on numbers. They include:
- Multiplication (often represented by ×)
- Division (often represented by ÷or /)
- Addition (represented by +)
- Subtraction (represented by -)
The order in which these operations should be performed in an expression is as follows: multiplications and divisions first, then additions and subtractions. For example: 8 –2 ×3 = 8 –6 = 2, not 6 ×3 = 18.
To avoid ambiguity, we often use parentheses (or brackets) which take precedence over all four basic operations. We could have written 8–(2×3) to make the expression clearer, although we will still get the same answer of 2.
Multiplication
There are several ways of indicating that two numbers are being multiplied. For example, suppose we wish to multiply the numbers 5 and 7. Some common notations are: 5 × 7, 5 . 7, 5 (7), and (5) 7.
An important property of multiplication is its symmetry; that is, reversing the order of the numbers results in the same answer. In our case, 5 × 7 = 7 × 5= 35. Also, if two numbers have the same sign and are multiplied together, the result will be positive; if they have different signs the result will be negative.
Division
There are several ways of indicating that two numbers are to be divided. For example, suppose we wish to divide the number 15 by the number 5. Some common notations for this are 15/5 and 15 ÷ 5.
The number to be divided (15) is called the dividend or numerator, and the number this is divided by (5) is called the divisor or denominator. The answer to the division is called the quotient.
Division does not have symmetry. If the order of the numbers is changed, the answer is affected. For example, in our case, 15/5 is not the same as 5/15. Instead, reversing the order of the numbers results in the inverse of the actual answer. If the signs of the numerator and denominator are the same, the quotient will be positive; if the signs are different, the quotient will be negative.
Addition and Subtraction
Addition does have symmetry. The order in which the numbers appear does not affect the result. For example, the expressions 2 + 3 and 3 + 2 are the same as they both give a result of 5.
Subtraction does not have symmetry. The order in which the numbers appear does affect the result. For example, the expression 8 – 3 is equal to 5, but 3 – 8 is equal to–5, not 5. If you reverse the order of the numbers, it results in the negative of the actual answer.
When performing multiple additions and subtractions, treat a + and – sign immediately next to each other as a – sign. Treat two – signs immediately next to each other as a + sign.
Fractions
A fraction is an expression that appears in the form a/b. In a/b, a is called the numerator, and b is called the denominator. The denominator can take on any value except zero.
The fraction can be proper or improper. A proper fraction is one in which the numerator is less than the denominator. An improper fraction has a numerator greater than the denominator. Just like numbers, the mathematical operations that are performed on fractions are addition, subtraction, multiplication, and division.
To add fractions with the same denominator, add together the two numerators to obtain the new numerator. The denominator remains the same. For example, adding 2/5 and 1/5 gives 3/5. To add fractions with different denominators, we change fractions with different denominators into fractions with identical denominators ("Lowest Common Denominator") before we can add the numerators. For example, 2/3 + 1/2 = 4/6 + 3/6 which gives 7/6. The rules for addition also apply to subtraction.
To multiply fractions, we multiply the numerators to get the new numerator and multiply the denominators to get the new denominator. For instance, 4/5 × 3/4 is equal to (4 × 3) / (5 × 4), or 12/20.
To divide one fraction by a second fraction, invert the second fraction, then multiply it by the first. Thus, every division problem can be changed into one of multiplication. For example, dividing 5/7 by 6/7 results in 5/7 × 7/6, which gives 5/6.
Percentages
Percentages are an important part of our everyday lives. Percentages provide a very handy way of writing fractions. Percentages can be compared more easily than fractions.
Shops advertise discounts on products. These discounts are percentages. For example, a shop may advertise "Up to 50% off marked prices." Financial institutions quote interest charged to the client on loans or interest paid for money invested as a percentage. For example, "Housing Loans -4.95% per annum for the first 12 months." Companies describe their success or failure as an increase or decrease in profit levels. For example, C- Company's profit is down by 15% for the last financial year. A salesperson may be given a commission as payment for selling goods. The commission can be a percentage of the sales made. For example, "20% commission on all sales."
Articles such as antiques or jewellery may increase in value as time goes by (appreciation). Items such as equipment and machinery usually decrease in value (depreciation). Both appreciation and depreciation are expressed in percentages. Now that we have looked at different uses of percentages, let us consider the following example that requires solving using percentages:
Example: A salesman is on a commission rate of 5%. How much commission do they make on sales worth $600,000?
Solution: 5/100 × 600,000. This gives a commission of $30,000.00.
Exponents
An exponent (or power) is a number that is written as a superscript to another number called the base. It tells how many times the base is to be multiplied by itself. The position of the superscript is to the right of and a little above the base. A number that is expressed as a base with an exponent is said to be in exponential form. This form is written as an where a is the base and n is the exponent or power.
Now, suppose that two numbers with the same base are expressed in exponential form:
- The product of those two numbers will have the same base with an exponent equal to the sum of the two original exponents. That is: am×an=am+n.
- The quotient of those two numbers will have the same base with an exponent equal to the difference between the exponents. That is: am÷an=am-n.
We also need to take note that a number expressed in exponential with a negative exponent is called the reciprocal (or inverse) of the same number with the negative sign removed from the exponent.
The definition of an exponent an can be extended to include situations where n is a fraction. In particular, if n is of the form 1 (where k is an integer), a1/k is said to be the "kth root of a." Under our definitions, the expression a0=an-n=1. Therefore, any number (except 0) raised to the exponent 0 is 1.
Algebraic Equations
Algebra is the study of the mathematical art of reasoning by means of a systematised notation, including letters and other symbols. For our purposes, the symbols used will be the letters of the alphabet. A function is an expression or relationship involving one or more variables. For example, 10w + 2t.
A variable is a term whose quantity can have different values. A constant is a term whose value does not change. One of the important uses of algebra is the solving of equations and inequalities.
We will look at techniques for solving linear and nonlinear equations and inequalities. An algebraic equation is thus a mathematical statement that relates two algebraic expressions involving at least one variable.
Some examples of equations with x as the variable are: 3x-2=7; 2x² –3x+5=0. The following are the properties of equality: For a, b, and c, any real numbers,
- If a=b, then a+c=b+c. Addition Property
- If a=b, then a-c=b-c. Subtraction Property
- If a=b, then ca=cb, c≠0. Multiplication Property
- If a=b, then, a/c=b/c, c≠0. Division Property
- If a=b, then either a or b may replace the other, Substitution Property, in any statement without changing the truth or falsity of the statement.
Linear Equations
Any equation that can be written in the form: ax + b = 0, a ≠ 0 Standard Form where a and b are real constants, and x is a variable, is called a linear, or first-degree equation in one variable. For example, 5x -9 = 3x + 7 and A = P + Prt are linear equations.
Linear equations define straight lines. The straight line (linear) equation ax + by = c can be drawn on a graph by using the following two facts (sometimes known as the axis-intersection ratio method):
- The line cuts the x-axis at x = c/a
- The line cuts the y-axis at y = c/b
For example, if you want to plot the line 5x + 10y = 200, this line crosses the y-axis at 200/10 (20) and the x-axis at 200/5 (40). The line can be drawn by joining these two points.
What if someone wants to solve linear equations? You can solve for any of the variables in a linear equation by making it (the variable) the subject of the formula. For example, solve for x and y in the following linear equations:
a) 4x-10 = 2x + 8
Solution: 4x – 2x = 8+10
2x = 18
x = 9
b) Z = 5y – by
Solution: y (5 – b) = Z
y = Z/(5-b)
Linear equations are sometimes expressed in word problems. For example, Find four consecutive even integers such that the sum of the first three exceeds the fourth by 8. In this case, you can solve the problem by following the 7 steps below:
Read the problem carefully—several times if necessary—that is, until you understand the problem, know what is to be found, and know what is given.
Let one of the unknown quantities be represented by a variable, say x, and try to represent all other unknown quantities in terms of x. This is an important step and must be done carefully.
- If appropriate, draw figures or diagrams and label known and unknown parts.
- Look for formulas connecting the known quantities to the unknown quantities.
- Form an equation relating the unknown quantities to the known quantities.
- Solve the equation and write answers to all questions asked in the problem.
- Check and interpret all solutions in terms of the original problem—not just the equation found in step 5—since a mistake may have been made in setting up the equation in step 5.