At bladeko, we offer concise online courses in algebra. These math algebra courses are designed to make your math degree easy. In this article, we are talking about algebra from A to Z.
1- Numbers chapter is the basis of most courses in Algebra
Natural numbers: The numbers well known to everyone are 0, 1, 2, 3, and so on. In lessons of algebra, these numbers are called natural (whole, or counting) numbers and are used to count objects like I have 3 apples, I have 5 sisters.
In all courses in Algebra, the set of all whole numbers is denoted $ {0,1,2,3, \cdots }; $ the dots mean that we are talking about an infinity of numbers. With whole numbers, we can do an addition, a subtraction of the object of similar characteristics, like 2 apples plus 3 apples equal to 5 apples (2 + 3 = 5). In a box of 50 oranges, if we take 10 oranges between them there are 40 oranges left in the box, this is exactly the subtraction of 10 oranges from 50 0ranges (50-10 = 40). On the other hand, we can multiply whole numbers; for example, if we want to determine the area of ​​a house: The area of ​​a house at 10 meters (width) and 12 meters (Length) is 10 x 12 = 120 meters. In general, we use multiplication to calculate areas and volumes.
negative whole numbers: In general, some teachers find this part difficult to learn in math for beginners. In this online algebra course, we will give a very simple introduction to such numbers.. these numbers are the opposite of whole numbers. In fact, the number $ 0 $ doesn’t mean anything, and all other natural numbers are greater than $ 0 $ (these are called positive whole numbers). In everyday life, we sometimes use numbers less than the number $0$. Let’s give an example to explain this situation, compare the temperature in Africa and Alaska, the temperature in Africa is always above $ 0$; it corresponds to a natural number. However, in Alaska the temperature is below $0$, sometimes it can be -40 degrees Celsius.
Any number that is less than zero is called a negative number. We represent these numbers as follows
Thus, positive numbers are placed to the right of zero, while negative numbers to the left of zero. Moreover, zero is a neutral number.
Integer numbers: are whole, and negative numbers. The zero is also an integer number (neutral number). The representation of integer numbers:
Division of numbers: Division is splitting objects into equal parts or groups. A bag of apples contains 12 apples and we want to divide this bag into 3 small bags so we divide 12 by 3, and therefore each bag must contain 4 apples. It’s division 12 by 4 and we write 12 ÷ 3 = 4 (we also write 12/3=4). The number 12 is called the dividend, the number 3 the divisor, and the 4 the quotient.
It’s a perfect division. However, in some situations, the division is perfect because you cannot divide a set of things into groups of the same number of items. For example, we have a bag that holds 8 oranges and we have 2 other small bags that can only hold 3 oranges. Each sachet should contain 3 oranges and 2 oranges remain. We, therefore, have a remainder number 2. We write 8 ÷ 2 = 3 R 2.
Rational numbers: Here we define more general numbers. A rational number is obtained by dividing two integers. An example:
For example $ \frac {1}{8} $ means $1$ part of $8$ parts. Imagine you bought a pizza that is divided into $8$ equal parts. If you eat 1 part there are $7$ parts of $8$ parts left, that is, you eat $ \frac{1}{8} $ of Pizza and there is $\frac{7}{8}$ left. This situation is represented in the following figure:
In the fraction $\frac{2}{5}$, the number $2$ is called the numerator, and the $5$ is called denominator
As with integers, we can also add, substitute and multiply fractions or rational numbers. The following rule shows the addition of fractions that have the same denominators. In this case, just add the numerators as shown in the following figure:
In most cases, the denominators are different. But no problem, we can always add substitute ads. The idea is to transform these fractions a bit into equivalent fractions with a common denominator. For example, if we want to add the fractions 1/2 and 1/3, we first find the common multiple of 2 and 3 (a number that divides 2 and 3) which is 6. Thus
Thus sum $\frac{1}{2}+\frac{1}{3}$ is equivalent to
The same calculation remains valid if the addition is replaced by the substitution.
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Online courses in algebra: Solving one-step equations
In pre-algebra, we can encounter the following problem: find the value of the unknown number $ x $ which satisfies the following equation (the equilibrium)\begin{align*}{\Huge ax+b=0},\end{align*}where $a$ and $b$ are known numbers with $a$ not equal to zero. This problem is called a one-step equation or one-variable equation. The number $ x $ is called the variable that we want to determine.
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To start solving the above equation, let’s remember that the equation is unchanged if we add the same number on both sides of the equations. So the first step to do is eliminate the number $ b $ from the left side of the equation. To do this, just add the opposite of $ b $ which is $ (- b) $ on both sides of the equation. We get \begin{align*} ax+b+(-b)=0+(-b).\end{align*}As $b+(-b)=0$ and $0+(-b)=-b,$ the one-step equation becomes \begin{align*}{\large ax=-b}.\end{align*} As the nomber $a$ is non null, we then divide the both sides of this equation by $a$, we obtain
\begin{align*} \frac{ax}{a}=\frac{-b}{a}.\end{align*} But $\frac{ax}{a}=\frac{a}{a} x=x$, then the solution of the equation is
\begin{align*}{\Large x=\frac{-b}{a}}.\end{align*}
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Examples: 1- The solution of the one-step equation $3x+2=0$ is $x=\frac{-3}{2}$.
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2- The solution of the equation $\frac{2}{3}x-4=0$ is
\begin{align*} x=\frac{4}{\frac{2}{3}}= 4 \frac{3}{2}=2 \times 3=6.\end{align*}
3- Let’s solve the equation $5x+12=2x+9$. In this case, we are using the rule $A=B$ is the same as $A-B=0$. So our equation can be rewritten as \begin{align*} 5x+12-(2x+9)=0.\end{align*} On the other hand, as $-(2x+9)=-2x-9$, then $ 5x+12-2x-9=0$. This means that $5x-2x+12-9=0$. So $3x+3=0$. Therefore, we have turned the problem into a standard one-set equation that we can easily solve, so the solution is $x=\frac{-3}{3}=-1$.
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The exponents of numbers (powers, subscript) are the most important part of all algebra lessons for beginners
Positive integer exponents:Â such numbers are used to simplify the expression of a number multiplied by itself many times. For example, if $k$ is an (integer or rational) number, we denote
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The whole number $n$ is called the exponent ( sometimes called the index, or the power), while the number $k$ is called the base. So the exponent is equal to the number of times the base number is multiplied by itself.
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We notice that $k^0=1$ and $k^1=k$.
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Now let $n$ and $m$ be two natural numbers and $k$ is an integer of a rational number. The product $k^ntimes k^m$ is given by
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The proof of the above formula is
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Let us now show what the multiplication of $k^m$ by itself in $n$ times
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The proof of this result is very simple. In fact, we have the following explication
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Negative integer exponents: such numbers are used to represent the powers of fractions. By definition, if $n$ is a positive integer and $k$ is an integer then
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From this definition we deduce that $k^n\times k^{-n}=1=k^{0}=k^{n-n}$.
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Exercise:Â Â Prove that if $m$ is a positive integer, then
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Solution: Observe that \begin{align*} \frac{b}{a}=\frac{1}{\frac{a}{b}}.\end{align*} Now by using the above definition, we get \begin{align*} \left( \frac{b}{a}\right)^m=\frac{1}{\left(\frac{a}{b}\right)^m}= \left( \frac{a}{b}\right)^{-m}.\end{align*}
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From the above discussion, for any integers $m$ and $n$ (positive or negative) and any number $k$ (integer of rational), we have \begin{align*}{\Huge k^{m+n}=k^nk^m.}\end{align*}
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Exercise: Calculate the following number
\begin{align*} {\Huge left(-3^{-2}\right)^{-2}}.\end{align*}
Solution: We never we have a negative power we need to transform it to a fraction. In fact, we have
\begin{align*}Â \left(-3^{-2}\right)^{-2}&= \frac{1}{(-3^{-2})^2}= \frac{1}{(-1)^2(3^{-2})^2}\cr &= \frac{1}{\left(\frac{1}{3^2}\right)^2}=\frac{1}{\frac{1}{3^4}}=3^4=81.\end{align*}
You need to practice more exponent exercises to familiarize yourself with the power of numbers.
