Positive numbers

Addition

See Tutorial:arithmetic

Failed to parse (syntax error): {\displaystyle 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 }

0	1	2	3	4	5	6	7	8	9	10	11
1	2	3	4	5	6	7	8	9	10	11	12
2	3	4	5	6	7	8	9	10	11	12	13
3	4	5	6	7	8	9	10	11	12	13	14
4	5	6	7	8	9	10	11	12	13	14	15
5	6	7	8	9	10	11	12	13	14	15	16
6	7	8	9	10	11	12	13	14	15	16	17
7	8	9	10	11	12	13	14	15	16	17	18
8	9	10	11	12	13	14	15	16	17	18	19
9	10	11	12	13	14	15	16	17	18	19	20
10	11	12	13	14	15	16	17	18	19	20	21
11	12	13	14	15	16	17	18	19	20	21	22

  ¹
  27
+ 59
————
  86

Multiplication

See Tutorial:multiplication

Failed to parse (syntax error): {\displaystyle 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 }

1 2 3 4 5 6 7 8 9 10 11 2 4 6 8 10 12 14 16 18 20 22 3 6 9 12 15 18 21 24 27 30 33 4 8 12 16 20 24 28 32 36 40 44 5 10 15 20 25 30 35 40 45 50 55 6 12 18 24 30 36 42 48 54 60 66 7 14 21 28 35 42 49 56 63 70 77 8 16 24 32 40 48 56 64 72 80 88 9 18 27 36 45 54 63 72 81 90 99 10 20 30 40 50 60 70 80 90 100 110 11 22 33 44 55 66 77 88 99 110 121

Notes Even numbers. Just add the number to itself. Sum of digits is a multiple of 3.   Last digit either 0 or 5.       First digit is easy. Sum of digits is 9. Just add a zero. Easy.

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

Powers

See Tutorial:exponents

Failed to parse (syntax error): {\displaystyle 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)} \\ }

${\displaystyle 5^2}$ ("five squared") is the area of a square that is 5 units wide.
${\displaystyle 5^3}$ ("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 numbers but you can easily work that out in your head by multiplying the number times itself.

From Wikipedia:Exponentiation:

n	n2	n3	n4	n5	n6	n7	n8	n9	n10
2	4	8	16	32	64	128	256	512	1,024
3	9	27	81	243	729	2,187	6,561	19,683	59,049
4	16	64	256	1,024	4,096	16,384	65,536	262,144	1,048,576
5	25	125	625	3,125	15,625	78,125	390,625	1,953,125	9,765,625
6	36	216	1,296	7,776	46,656	279,936	1,679,616	10,077,696	60,466,176
7	49	343	2,401	16,807	117,649	823,543	5,764,801	40,353,607	282,475,249
8	64	512	4,096	32,768	262,144	2,097,152	16,777,216	134,217,728	1,073,741,824
9	81	729	6,561	59,049	531,441	4,782,969	43,046,721	387,420,489	3,486,784,401
10	100	1,000	10,000	100,000	1,000,000	10,000,000	100,000,000	1,000,000,000	10,000,000,000

Negative numbers

Subtraction

Subtraction is the opposite of addition.

${\displaystyle 5-3=2\quad}$ because ${\displaystyle \quad2+3=5}$

But then what is

${\displaystyle 3-5=?}$

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

0	1	2	3	4	5	6	7	8	9
1	0	-1	-2	-3	-4	-5	-6	-7	-8
2	1	0	-1	-2	-3	-4	-5	-6	-7
3	2	1	0	-1	-2	-3	-4	-5	-6
4	3	2	1	0	-1	-2	-3	-4	-5
5	4	3	2	1	0	-1	-2	-3	-4
6	5	4	3	2	1	0	-1	-2	-3
7	6	5	4	3	2	1	0	-1	-2
8	7	6	5	4	3	2	1	0	-1
9	8	7	6	5	4	3	2	1	0

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 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

Powers with negative numbers

A negative number raised to an even number is positive

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

and a negative number raised to an odd number is negative

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

A number raised to a negative number is a fraction

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

and

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

Therefore

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

Fractions

Division

See Tutorial:fractions

Division is the opposite of multiplication.

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

and

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

and

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

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

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

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.

Failed to parse (syntax error): {\displaystyle 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 }

Failed to parse (syntax error): {\displaystyle 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 }

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.

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

and

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

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

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

and

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

Further reading

• Introductory mathematics

References