**Numbers, Calculations and Algebra**

Quiz yourself

Algebra

Algebra is a part in mathematics where letters are usually used to represent other numbers.

Numbers and Calculations

Numbers can be divided into many sub categories. We will be learning about natural and whole numbers, rational and irrational numbers, real numbers and integers.

Variables, Constants, Terms and Expressions

Variable

- letter or symbol used in place of a number. ex. y

Constant

- number with a fixed value. ex. 7

Term

- can be just a variable or a constant or a combination of variable and constant using multiplication or division. ex. y, 7, 7y, 7/y

Expression

- combination of two or more terms using addition or subtraction. ex. 7y - 3

Look at the chart for more information on the different categories of numbers

Operations with integers

Addition and Subtraction

Multiplication and Division

Multiplication and division of integers is the same as multiplying and dividing positive numbers, but the number of negative decides if the answer will be positive or negative.

When all the numbers are positive, the product, or quotient will also be positive.

When there are an odd number of negative signs, the answer will be negative.

When there are an even number of negative signs, the answer will be positive.

Always remember

(-)*(-) and (+)*(+) are equal to +, and

(-)*(+) is equal to -

And simplify the signs before trying to solve the equation

ex. 9 -(-5) = 9 + 5

When adding or subtracting integers, first simplify the signs then add or subtract. After simplifying the signs, if you end up with something like 4 - 9 or -9 + 4 you should follow the steps listed below.

Look for the larger number ignoring the signs. In this case 9. The answer will carry the same sign as the larger number. - in this example.

Then, subtract the smaller number from the larger number, and place the difference after the sign to get your answer. -5 is the answer here.

After simplifying, if you end up with something like -3 - 6 or 4 + 5, follow the steps below.

Add the numbers without their signs.

Then add a + sign if the numbers are positive, or a - sign if the numbers are negative.

Try out some problems for yourself!

9 +(-6) =

-3 - 7 =

4 -(-7) =

-7 -(-6) =

-7 -(-3) +(-4) =

Try out some equations for yourself!

-7 * -3 =

-4 * 9 =

-8 * -12 =

17 * -14 =

9 * -7 + 5 =

Squares, Cubes, and Exponents

When we want to represent a number multiplied by itself for a certain number of times, we can use exponents to do so.

For ex. 7 * 7 can be represented as 7^2.

The number two here is the exponent.

This is read as seven to the second power or seven to the power of two, or seven squared.

A perfect square is a number that can be made by multiplying two numbers of equal value together.

For ex. 49, 64, 169

A number that can be made by multiplying a number by itself three times is called a perfect cube.

For ex. 7^3 (seven to the power of three) is equal to 343.

Try out these questions!

Represent 7 * 7 * 7 in exponential form

What is 9^3 equal to?

What is 5 squared equal to?

10 to which power is equal to 1000?

Why are exponents used?

Square Roots and Cube Roots

A square root will produce another number when multiplied by itself.

The 64^1/2 (square root of 64) is equal to 8 because 8 * 8 = 64

A cube root is similar to a square root, but will produce another number when multiplied by itself thrice.

Square roots are the reciprocals of squares, and cube roots are the reciprocals of cubes.

Finding Square and Cube Roots

When finding square roots or cube roots, first find the prime factorization of the number and see if any of the prime factors appear more than once. This will help in solving these kind of problems.

For ex.

square root of 64 = 8

The prime factorization of 64 is 2^6

which is same as

square root of 64 = square root of 2^6

sq root of 64 = sq root of 2^2 . sq root 2^2 . sq root 2^2

sq root of 64 = 2 . 2 .2

sq root of 64 = 8

Now find the square root of 1296

Types of Expressions and Simplifying Expressions

Types of expressions:

Monomial Expressions- contains one term

Binomial Expressions- contains two terms

Trinomial Expressions- contains three terms

Polynomial Expression- contains more than one term

Simplifying Expressions

Refers to the grouping of like terms (those with same variable).

For ex. 10x and 12x are like terms.

An expression can be simplified by adding or subtracting like terms together.

For ex. 10x + 12x can be simplified into 22x.

Test your understanding

What type of expression is 3x + 9y - 7r?

Simplify the expression 7x - 3x + 8y

Are 7x and 7y like or unlike terms?

Functions

Function is a formula which gives an output for an input

The function for an unknown variable can be written as f(x) = "the formula"

This is read as f of x is equal to "the formula"

Ex. f(x) = 7x + 3

x can be substituted for any number.

If x = 3, the formula can be written as f(3) = 7(3) + 3

Try out these problems!

Is f(y) = 30y - 4 a function?

If x=6, solve f(x) = 3x + 2

Equations

An equation consists of two expressions separated by an equal sign.

The expression on the left of the equal sign must be equal to the expression on the right.

When there is a variable in the equation, you have to solve for the variable

To do this, first the expressions should be simplified. Then all the variables should be brought over to one side of the equation. Finally, the variable should be isolated.

Answer the following questions

Is 7x - 3 + 4x an equation?

Solve for x in the following equations:

8x -2 = 6x + 2

9x + 3 = 4x + 8

1. Solve -9 * 3

2. Solve -9 - 3

3. What is 9^3

4. Find the square root of 121

5. Find the cube root of 729

6. Solve for x

a. 5x + 3 = 13x/2

b. 10x - 3 = 18x - 27

7. Simplify the expressions

a. 10y - 3y + 8x + 3x

b. 7r + 8r - 6p

8. Find the output of this function when x is equal to 4

a. f(x) = 7x - 9

b. f(x) = -4x + 7