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# Algebraic Notation

Did you ever wonder how real-life problems could be sometimes solved mathematically? Did you wonder how these problems could be expressed so concisely that the right solutions were found using mathematics?

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In this explanation, we will discuss algebraic notation, symbols that are involved, examples, and formulas that are born from algebraic notation.

## What is algebraic notation?

Algebraic notation is the representation of mathematical concepts and ideas in a concise manner. This employs the use of letters to represent unknown values in real-life problems.

The fundamentals of algebra are built around symbols that are used to represent the problems that require mathematical principles to solve, and the relationship between them. Letters such as a, b, c, x, y, and z are the most common variables to come across, that usually represent quantities. Lowercase Greek letters such as α, β, γ, θ are habitually used to represent angles and planes. They are then put into statements with the help of mathematical operators that denote the relationship between them.

## Examples of algebraic notation

As discussed earlier, algebraic notation is used to make express statements more concisely.

• To write repeated multiplication of values, use a superscript to represent how many times you want to multiply itself. For example, $a×a×a$$a×a×a$ is represented as ${a}^{3}$${a}^{3}$, where $a$ here represents any number.

• The Greek letter Pi, π, represents a special number 3.14159...

• The equal sign, =, is used to declare that two quantities are equal.

• The less than <, less than or equal to, ≤, greater than, >, greater than or equal to, ≥, and not equal to, ≠, signs are all symbols that are also used to denote the inequality of quantities.

• The operators +, -, ×, and ÷ are used to denote the relationship between values such that if you have $2x+1$, it means you need to add 2x to 1.

### What are variables?

Variables are letters that are used to represent unknown quantities in algebra.

For example, in the expression $2x+1$, 'x' is the variable as it is the letter here. The number that multiplies the variable is known as the coefficient. Hence, 2 is the coefficient in this expression. Whilst 1 in this expression is the constant or independent term, since its value does not depend on the variable x.

Knowing that the variables represent unknown numbers, is usually what the problem is built around. Hence, finding the solution to such problems means we need to find the value of the variable.

For example, if Mike purchased a pair of shoes and a shirt for £100 but he knows only the cost of the pair of shoes to be £70. How much does the shirt cost?

This could be modelled into a mathematical statement where the cost of the shirt can be represented by the letter x.

$£70+x=£100$

To find the value of x, we must solve the equation above. We will isolate x such that x is the only element that remains on one side of the equation whilst everything else remains on the other side.

By that, we will add minus £70 from each side of the equation.

$£70-£70+x=£100-£70x=£30$

This here means that the value of x is £30. Hence the cost of the shirt is £30.

In other cases, variables can be useful in evaluating mathematical statements.

We can take the same example of Mike's purchase. Before finding the solution, he could assume the shirt cost him £35. However, he could use substitution to help evaluate the proposed solution he has. All we need to do is to substitute the proposed value into the place of the variable and see if it satisfies the equation.

$£70+x=£100£70+£35=£100£105=£100$

We realise here that the statement suggests that £105 is the same as £100, which is not true. So it is safe to conclude that the proposed solution by Mike that the shirt cost him £35 is wrong.

## Algebraic notation symbols

The most commonly used symbols in algebra are listed in the table below.

 Symbol Meaning Example + Add $4+7=11$ - Subtract $13-7=6$ × Multiply $3×4=12$ ÷ Divide $25÷5=5$ √ Square root $\sqrt{36}=6$ $\text{exception 25:}$$\sqrt[3]{}$ Cube root $\sqrt[3]{27}=3$ $\sqrt[n]{}$ nth root $\sqrt[5]{32}=2$ ( ) Grouping symbols $2\left(x+1\right)$ [ ] Grouping symbols $21+\left[4\left(2+1\right)+3\right]$ { } Set symbols $A=\left\{1,2,3,4,5\right\}$ = Equals $2x+1=12$ ≠ Not equal to $3+4\ne 13$ < Less than $3x<7$ ≤ less than or equal to $3x\le 7$ > Greater than $5-1>4x$ ≥ Greater than or equal to $5-1\ge 4x$ $⇒$ Implies $aandbareodd⇒a+biseven$ ∴ Therefore $a=b\therefore b=a$

## Sigma notation and a few other

In algebra, a lot more symbols are used for specific cases. We are going to explore in this section what the sigma notation, the Pi notation, and the factorial are.

### Sigma notation

The sigma notation is the most convenient way to express long sums.

For example, $1+2+3+4+5$ could also be written as

id="2943293" role="math" $\sum _{i=1}^{5}i$.

This means that we are summing up all the values of $i$ starting from $i=1$ until we get to $i=5$, which is where we stop.

${3}^{2}+{4}^{2}+{5}^{2}+{6}^{2}+{7}^{2}+{8}^{2}+{9}^{2}+{10}^{2}=\sum _{n=3}^{10}{n}^{2}$

Notice that plugging in the values of n should give you the answer you are looking for.

### Pi notation

Pi notation is used to denote repeated multiplication. It is also called product notation. This notation is quite similar to summation notation. An example is given below.

$\prod _{n=5}^{N}\left({n}^{2}-1\right)=\left({5}^{2}-1\right)\left({6}^{2}-1\right).....\left({N}^{2}-1\right)$

This reads the products from $n=5$ to N, where N is larger than n.

Pi notation is also used to define the factorial $n!$

$n!=\prod _{i=1}^{n}i=\left(1\right)\left(2\right)\left(3\right)\left(4\right)....\left(n-1\right)\left(n\right)$

### Counting notation

You are likely to come across the notation $n!$. This represents the factorial;

$n!=1$ if $n=0$,

otherwise,

$n!=n×\left(n-1\right)×\left(n-2\right)×\left(n-3\right)×...×3×2×1$.

$n!$ counts the number of ways to arrange n distinct objects. So it is intuitive to know that when you have zero (0) objects, there is only one way to arrange them – do nothing.

Related to factorials is the binomial coefficient notation $\left(\begin{array}{c}n\\ k\end{array}\right)$.

$\left(\begin{array}{c}n\\ k\end{array}\right)={}^{n}C_{k}=\frac{n!}{\left(n-k\right)!k!}$

The formula above is a way to express the number of k subsets in an n set. So here we think of n as a non-negative integer and k as a non-negative integer that is less than or equal to n.

## Formulas as algebraic notation

Mathematical formulas are typical expressions of algebraic notations. Formulas consist of different quantities connected together with the equal sign. They contain variables and sometimes constants. This means that if you have the values of certain variables in a formula you can find the value of the remaining variables, the same properties of equations.

Below are examples of mathematical formulas that are typical expressions of algebraic notation.

 Concept Formula Area of rectangle $A=l×w$ Area of circle $A=\pi {r}^{2}$ Volume of cuboid $Volume=l×b×h$ Volume of cylinder $Volume=\pi {r}^{2}\mathrm{h}$ Speed $Speed=\frac{Dis\mathrm{tan}ce}{Time}$ Density $Density=\frac{Mass}{Volume}$

## Algebraic notation method

The algebraic notation method is a way to use expansion to multiply large numbers. Let us look at the steps taken to perform operations in this manner.

• STEP 1: Expand the given values

• STEP 2: Use the distributive property to perform multiplication operations.

Let us try cases where we multiply a number with a single place value by a number with two place values.

Use the algebraic notation method to perform the multiplication;

$4×91$

Solution

First, we will write 91 in expanded form.

$4×91$

91 can be expanded as $90+1$

$4×\left(90+1\right)$

We can now use the distributive property here to multiply the values. 4 will now multiply both given values.

$\left(4×90\right)+\left(4+1\right)==360+4=364$

Use the algebraic notation method to perform the multiplication;

$5×43$

Solution:

We will first write 43 in expanded form. 43 can be expanded as $40+3$

$5×\left(40+3\right)$

We can now use the distributive property here to multiply the values.

$\left(5×40\right)+\left(5×3\right)==200+15=215$

Let us also try cases where we multiply that have two place values.

Use the algebraic notation method to perform the multiplication;

$23×42$

Solution:

Here, we will write both numbers in expanded form. 23 can be written as $20+3$, and 42 can be written as $40+2$

$\left(20+3\right)×\left(40+2\right)$

We will now use the distributive property to perform the operation

$20\left(40+2\right)+3\left(40+2\right)==800+40+120+6=966$

Use the algebraic notation method to perform the multiplication;

$75×13$

Solution:

We will write both numbers in expanded form. 75 will be $70+5$ and 13 will be$10+3$.

$\left(70+5\right)×\left(10+3\right)$

We will now use the distributive property to perform the operation.

$70\left(10+3\right)+5\left(10+3\right)==700+210+50+15=975$

## Algebraic Notation - Key takeaways

• Algebraic notation is the representation of mathematical concepts and ideas in a concise manner.
• Letters such as a, b, c, x, y, and z are the most common variables to come across.
• Variables are letters that are used to represent unknown quantities in algebra.
• To write repeated multiplication of values, use a superscript to represent how many times you want to multiply itself.
• The operators +, -, ×, and ÷ are used to denote the relationship between values.
• The less than <, less than or equal to ≤, greater than >, greater than or equal to ≥, and not equal to ≠ signs are all symbols that are also used to denote the inequality of quantities.

#### Flashcards in Algebraic Notation 5

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How do you write algebraic notation?

Letters are used to represent unknown quantities.

What is algebraic notation?

Algebraic notation is the representation of mathematical concepts and ideas in a concise manner.

What is interval notation in algebra?

Interval notation is a way to write sets of continuous elements, like real numbers, using the numbers that bound them.

How to write a domain in algebraic notation?

The domain of a function is the set of all possible inputs of the function. The domain of a function f(x) is given as Domain(f) = {x, x∈A}, where A is a subset of a number set.

What is set notation in algebra?

Set notation is used in mathematics to essentially list numbers, objects, or outcomes.

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