Elementary mathematics

This article is Grade school level

5
0

=

0
5

=

5

5
1

=

1
5

=

6

5
2

=

2
5

=

7

5
3

=

3
5

=

8

5
4

=

4
5

=

9

5
5

=

5
5

=

10

{\displaystyle
\begin{align}
& 5+0 \quad = \quad 0+5 \quad = \quad 5 \\
& 5+1 \quad = \quad 1+5 \quad = \quad 6 \\
& 5+2 \quad = \quad 2+5 \quad = \quad 7 \\
& 5+3 \quad = \quad 3+5 \quad = \quad 8 \\
& 5+4 \quad = \quad 4+5 \quad = \quad 9 \\
& 5+5 \quad = \quad 5+5 \quad = \quad 10
\end{align}
}

```
  ¹
  27
+ 59
————
  86

```

5
−
# 3
2

{\displaystyle 5-3=2\quad}

 because

2
# 3
5

{\displaystyle \quad2+3=5}

But then what is

3
−
# 5
?

{\displaystyle 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 Wikipedia:Negative number

### Addition with negative numbers

Adding a negative number is the same as subtracting a positive number.

(5) + (−3)  = 5 − 3 = 2.
Addition of two negative numbers is very similar to addition of two positive numbers. For example,

(−5) + (−3)  =  −8.
### Subtraction with negative numbers

Subtracting a negative number is the same as adding a positive number.

 5 − (−3)  =  5 + 3  =  8
and

 (−5) − (−8)  =  (−5) + 8  =  3.
## Multiplication

See Tutorial:multiplication

5
⋅
0

=

0
⋅
5

=

0

5
⋅
1

=

1
⋅
5

=

5

5
⋅
2

=

2
⋅
5

=

5
5

5
⋅
3

=

3
⋅
5

=

5
5
5

5
⋅
4

=

4
⋅
5

=

5
5
5
5

{\displaystyle
\begin{align}
& 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 \end{align}
}

```
         2 4
       × 3 7

       1 6 8
   +   7 2

   =   8 8 8

```

### 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

 (2) × (3)  =  6
and
 (−2) × (−3)  =  6.
If the two numbers have different signs, the result is always negative

 (−2) × (3)  =  −6
and
 (2) × (−3)  =  −6
## Fractions

### Division

See Tutorial:fractions
Division is the opposite of multiplication.

6
3

=
2

{\displaystyle \frac{6}{3} = 2 \quad}

 because

2
⋅
# 3
6

{\displaystyle \quad 2 \cdot 3 = 6}

and

2
5

⋅

3
4

=

2
⋅
3

5
⋅
4

=

6
20

=

3
10

⋅

2
2

=

3
10

⋅
# 1

3
10

{\displaystyle \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

500
4

=
125

{\displaystyle \frac{500}{4}=125}

this can be worked out by hand like this

```
     125      (Explanations)
   4)500
     4        ( 4 ×  1 =  4)
     10       ( 5 -  4 =  1)
      8       ( 4 ×  2 =  8)
      20      (10 -  8 =  2)
      20      ( 4 ×  5 = 20)
       0      (20 - 20 =  0)

```

But then what is

5
4

=
?

{\displaystyle \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 Wikipedia:Long division:

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

```
     1.25      (Explanations)
   4)5.00
     4         ( 4 ×  1 =  4)
     1.0       ( 5 -  4 =  1)
       8       ( 4 ×  2 =  8)
       20      (10 -  8 =  2)
       20      ( 4 ×  5 = 20)
        0      (20 - 20 =  0)

```

The "." is called the Decimal point.

```
     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 × 4 = 28)
         20     (an additional zero is brought down)
         20     (5 × 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.

1

/

# 1
1.0

1

/

# 2
0.5

1

/

# 3
0.333

333
¯

1

/

# 4
0.25

1

/

# 5
0.2

1

/

# 6
0.1666

666
¯

1

/

# 7
0.142857

142857
¯

1

/

# 8
0.125

1

/

# 9
0.111

111
¯

1

/

# 10
0.1

{\displaystyle
\begin{align}
1 / 1 = 1.0 \\
1 / 2 = 0.5 \\
1 / 3 = 0.333\overline{333} \\
1 / 4 = 0.25 \\
1 / 5 = 0.2 \\
1 / 6 = 0.1666\overline{666} \\
1 / 7 = 0.142857\overline{142857} \\
1 / 8 = 0.125 \\
1 / 9 = 0.111\overline{111} \\
1 / 10 = 0.1
\end{align}
}

2

/

# 1
2.0

2

/

# 2
1.0

2

/

# 3
0.666

666
¯

2

/

# 4
0.5

2

/

# 5
0.4

2

/

# 6
0.333

333
¯

2

/

# 7
0.285714

285714
¯

2

/

# 8
0.25

2

/

# 9
0.222

222
¯

2

/

# 10
0.2

{\displaystyle
\begin{align}
2 / 1 = 2.0 \\
2 / 2 = 1.0 \\
2 / 3 = 0.666\overline{666} \\
2 / 4 = 0.5 \\
2 / 5 = 0.4 \\
2 / 6 = 0.333\overline{333} \\
2 / 7 = 0.285714\overline{285714}\\
2 / 8 = 0.25 \\
2 / 9 = 0.222\overline{222} \\
2 / 10 = 0.2
\end{align}
}

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.

8
2

=
4

{\displaystyle
\frac{8}{2}=4
}

and

−
8

−
2

=
4

{\displaystyle \frac{-8}{-2}=4}

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

8

−
2

=
−
4

{\displaystyle
\frac{8}{-2}=-4
}

and

−
8

2

=
−
4

{\displaystyle
\frac{-8}{2}=-4
}

## Powers

See Tutorial:exponents

5
0

=
1

5
1

=
5

5
2

=
5
⋅
5

5
3

=
5
⋅
5
⋅
5

5
4

=
5
⋅
5
⋅
5
⋅
5

5
5

=
5
⋅
5
⋅
5
⋅
5
⋅
5

5
2

⋅

5
3

=
(
5
⋅
5
)
⋅
(
5
⋅
5
⋅
5
# )

5
5

=

5

(
2
3
)

{\displaystyle
\begin{align}
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)}
\end{align}
}

5
2

{\displaystyle 5^2}

 ("five squared") is the area of a square that is 5 units wide.

5
3

{\displaystyle 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 small numbers but you can easily work that out in your head by multiplying the number times itself.

From Wikipedia:Exponentiation:

### Powers of larger numbers

31
2

=
31

⋅

31

=
(
3
∗
10

1
)

⋅

(
3
∗
10

1
)

=
3
∗
10

⋅

3
∗
10

1

⋅

3
∗
10

3
∗
10

⋅

1

1

⋅

1

=

3
2

⋅

10
2

2

⋅

1

⋅

3
∗
10

1
2

=
900

60

1

{\displaystyle
\begin{array}{llll}
31^2 &= 31 & \cdot \quad 31 \\
&= (3 \* 10 \quad + \quad 1) &\cdot \quad (3 \* 10 \quad + \quad 1) \\
&= 3 \* 10 \, \cdot \, 3 \* 10 \quad &+ \quad 1 \, \cdot \, 3 \* 10 \quad &+ \quad3 \* 10 \, \cdot \, 1 \quad &+ \quad 1 \, \cdot \, 1 \\
&= {\color{red} 3^2 \, \cdot \, 10^2 \quad} & {\color{red} + \quad 2 \, \cdot \, 1 \, \cdot \, 3 \* 10 \quad } && {\color{red} + \quad 1^2 } \\
&= 900 \quad &+ \quad 60 \quad &+ \quad 1
\end{array}
}

### Powers with negative numbers

A negative number raised to an even number is positive

−

2
2

=
−
2
⋅
−
# 2
4

{\displaystyle -2^2=-2 \cdot -2 = 4}

and a negative number raised to an odd number is negative

−

2
3

=
−
2
⋅
−
2
⋅
−
# 2
−
8

{\displaystyle -2^3=-2 \cdot -2 \cdot -2 = -8}

A number raised to a negative number is a fraction

10
2

=
100

10
1

=
10

10
0

=
1

10

−
1

=

1
10

{\displaystyle
\begin{align}
10^2 = 100 \\
10^1 = 10 \\
10^0 = 1 \\
10^{-1} = \frac{1}{10}
\end{align}
}

and

2

−
1

=

1
2

{\displaystyle 2^{-1} = \frac{1}{2}}

and

2

−
2

=

1

2
⋅
2

=

1
4

{\displaystyle 2^{-2}= \frac{1}{2 \cdot 2} = \frac{1}{4}}

Therefore

5

3

⋅

5

−
3

=

5
⋅
5
⋅
5

5
⋅
5
⋅
5

=
# 1

5
0

=

5

(
3
−
3
)

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

## Order of operations

* **P** = evaluate **P**arentheses

* **E** = evaluate **E**xponents

* **M** = **M**utiply

* **D** = **D**ivide

* **A** = **A**dd

* **S** = **S**ubtract

But when the operations are only Add & Subtract or Multiply & divide, it is evaluated form left to right.

10 − 3 + 2 = 9
The correct value is 9 (not 5, as would be the case if you added the 3 and the 2 before subtracting from the 10).

10 − (3 + 2) = 5