Elementary mathematics

Addition

 * See

$$ 5+0\quad=\quad0+5\quad=\quad5 \\ 5+1\quad=\quad1+5\quad=\quad6 \\ 5+2\quad=\quad2+5\quad=\quad7 \\ 5+3\quad=\quad3+5\quad=\quad8 \\ 5+4\quad=\quad4+5\quad=\quad9 \\ 5+5\quad=\quad5+5\quad=\quad10 $$

¹  27 + 59 ————   86

Multiplication

 * See

$$ 5 \cdot 0 \quad = \quad 0 \cdot 5 \quad = \quad 0 \\ 5 \cdot 1 \quad = \quad 1 \cdot 5 \quad = \quad 5 \\ 5 \cdot 2 \quad = \quad 2 \cdot 5 \quad = \quad 5 + 5 \\ 5 \cdot 3 \quad = \quad 3 \cdot 5 \quad = \quad 5 + 5 + 5 \\ 5 \cdot 4 \quad = \quad 4 \cdot 5 \quad = \quad 5 + 5 + 5 + 5 $$

2 4             × 3 7             --        1 6 8        +   7 2     --    =   8 8 8

Powers

 * See

$$ 5^0 = 1 \\ 5^1 = 5  \\ 5^2 = 5 \cdot 5 \\ 5^3 = 5 \cdot 5 \cdot 5 \\ 5^4 = 5 \cdot 5 \cdot 5 \cdot 5 \\ 5^5 = 5 \cdot 5 \cdot 5 \cdot 5 \cdot 5\\ 5^2 \cdot 5^3 = (5 \cdot 5) \cdot (5 \cdot 5 \cdot 5) = 5^5 = 5^{(2+3)} \\ $$

$$5^2$$ ("five squared") is the area of a square that is 5 units wide. $$5^3$$ ("five cubed") is the volume of a cube that is 5 units wide.

As you can see in the following table, raising numbers to powers results in very large numbers.

You will not need to memorize this table. It will occasionally be useful to know the squares of numbers but you can easily work that out in your head by multiplying the number times itself.

From Exponentiation:

Subtraction
Subtraction is the opposite of addition.


 * $$5-3=2\quad$$ because $$\quad2+3=5$$

But then what is


 * $$3-5=?$$

What number when added to 5 results in 3? The answer is that there is no answer. So we have to create one.

We therefore create what are called negative numbers

From Negative number

Addition with negative numbers
Adding a negative number is the same as subtracting a positive number.



Addition of two negative numbers is very similar to addition of two positive numbers. For example,



Subtraction with negative numbers
Subtracting a negative number is the same as adding a positive number.



and



Multiplication with negative numbers
When multiplying numbers, the sign of the product is determined by the following rules:

If two numbers have the same sign, the result is always positive
 * and
 * and

If the two numbers have different signs, the result is always negative
 * and
 * and

Powers with negative numbers
A negative number raised to an even number is positive


 * $$-2^2=-2 \cdot -2 = 4$$


 * and a negative number raised to an odd number is negative


 * $$-2^3=-2 \cdot -2 \cdot -2 = -8$$

A number raised to a negative number is a fraction


 * $$2^{-1} = \frac{1}{2}$$


 * and


 * $$2^{-2}= \frac{1}{2 \cdot 2} = \frac{1}{4}$$

Therefore


 * $$5^{3} \cdot 5^{-3} = \frac{5 \cdot 5 \cdot 5}{5 \cdot 5 \cdot 5} = 1 = 5^0 = 5^{(3-3)}$$

Division

 * See

Division is the opposite of multiplication.


 * $$\frac{6}{3} = 2 \quad$$ because $$\quad 2 \cdot 3 = 6$$


 * and


 * $$\frac{2}{5} \cdot \frac{3}{4}=\frac{2 \cdot 3}{5 \cdot 4} = \frac{6}{20} = \frac{3}{10} \cdot \frac{2}{2}=\frac{3}{10} \cdot 1 = \frac{3}{10}$$


 * and


 * $$\frac{500}{4}=125$$


 * this can be worked out by hand like this

1 2 5      (Explanations) 4)500     4        ( 4 &times;  1 =  4)      1 0       ( 5 -  4 =  1 )       8       ( 4 &times;  2 =  8)       2 0      (10 -  8 =  2 )       20      ( 4 &times;  5 = 20)        0      (20 - 20 =  0)

But then what is


 * $$\frac{5}{4}=?$$

The answer is that there is no answer. So we have to create one.

So we create what are called decimal numbers.

From Long division:

An example is shown below, representing the division of 5 by 4, with a result of 1.25 ("one point two five").

1 . 2 5      (Explanations) 4)5.00     4         ( 4 &times;  1 =  4)      1 .0       ( 5 -  4 =  1 )        8       ( 4 &times;  2 =  8)        2 0      (10 -  8 =  2 )        20      ( 4 &times;  5 = 20)         0      (20 - 20 =  0)

The "." is called the.

31.75        4)127.00      12         (12 ÷ 4 = 3)       07        (0 remainder, bring down next figure)        4        (7 ÷ 4 = 1 r 3 )                                                     3.0      (0 is added in order to make 3 divisible by 4)        2.8      (7 &times; 4 = 28)          20     (an additional zero is brought down)          20     (5 &times; 4 = 20)           0

Some decimal numbers never end

0.333333333333333333333...     3)1.000000000000000000000...        9                1.0                9                 10                9                 10                  9                  10                   9                   10

The "..." at the end means that it just keeps repeating forever. This can also be indicated with an overline.

The sign rules for division are the same as for multiplication. For example,

If dividend and divisor have the same sign, the result is always positive.



\phantom{-}\frac{8}{2}=\phantom{-}4 $$


 * and


 * $$\frac{-8}{-2}=\phantom{-}4$$

If dividend and divisor have different signs, the result is always negative.



\frac{\phantom{-}8}{-2}=-4 $$


 * and



\frac{-8}{\phantom{-}2}=-4 $$