Algebraic Notation

Algebraic notation is a way of expressing mathematical ideas concisely. In algebra, letters (a, b, c, d, …, x, y, z,∝, β, etc) are used to represent unknown qualities.

7 + 7 + 7 + 7 + 7 is the same as 5 \times 7. In algebra, a + a + a + a + a is 5 \times a or just 5a.

{ b }^{ 2 } + { b }^{ 2 } + { b }^{ 2 } is 3{ b }^{ 2 }, since we have three { b }^{ 2 } added together.

2a stands for a+a and 3b stands for b+b+b.

a^{2} stands for a \times a, a^{3} stands for a \times a \times a.

a \times b is written simply as ab without the multiplication sign in the middle. A \times b \times c = Abc.

5a + 2a + 5b + 4ab + 4a - a + 7b - ab can be simplified to (5a + 2a + 4a - a) + (5b + 7b) + (4ab - ab) to give 10a + 12b + 3ab

Remember that 1a is the same as a, 1b is the same as b, 1abc is the same as abc, etc.

Usually, in an equation you could see 6y, which means 6 \times y or ‘6 multiplied by y’. We leave out the multiplication sign to make the equations look more elegant and neater when dealing with them. The number that is used to multiply a number is called the coefficient.

In the expression 10a + 12b + 3 ab - { b }^{ 2 }:

  • 10 is the coefficient of a
  • 12 is the coefficient of b
  • 3 is the coefficient of ab
  • -1 is the coefficient of { b }^{ 2 }

It is also important to note that fractions are another way to express division. For example, a \div b is expressed as \frac { a }{ b }

Let’s look at some examples

Example

Simplify the following

i) a + 2a + 3a +4a + 5a

ii) 7b - b + 3b - 2b + 4b

iii) 3ab + 5ab - 2ab

iv) 100{ x }^{ 2 } - { x }^{ 2 }

v) a + 2a + 3a + b + 3b + 5b - c + 3c

i) a + 2a + 3a + 4a + 5a = 15a

ii) 7b - b + 3b - 2b + 4b = 11b

iii) 3ab + 5ab - 2ab = 6ab

iv) 100{ x }^{ 2 } - { x }^{ 2 } = 99{ x }^{ 2 }

v) a + 2a + 3a + b + 3b + 5b - c + 3c = 6a + 9b + 2c


Example

Simplify a + 3a + 4b - 5c + 3b + 5a - 8b + c

a + 3a + 4b - 5c + 3b + 5a - 8b + c

We group the a’s together, the b’s together and the c’s together. This give us:

a + 3a + 5a + 4b + 3b - 8b - 5c + c

= 9a - b - 4c


Example

Simplify 4a + 7a - { a }^{ 2 } + 3ab - a + { 5a }^{ 2 } - 10ab

The a’s give us 4a + 7a - a = 10a

The a^{2}‘s give us -a^{2} + 5a^{2} = 4a^{2}

The ab’s give us 3ab - 10ab = -7ab

So, the simplified form of 4a + 7a - { a }^{ 2 } + 3ab - a + { 5a }^{ 2 } - 10ab is:

10a + 4a^{2} + \left( -7ab\right)

= 10a + 4a^{2} - 7ab

Note that we can also write:

  • { 4a }^{ 2 } + 10a - 7ab

or

  • { 4a }^{ 2 } - 7ab + 10a

or

  • -7ab + 10a + { 4a }^{ 2 }

They are all correct, since a + b = b + a.