Arithmetic operations on real numbers.
| Operation
| Symbols used
| Result's name
| Notes
|
| addition
| a + b
| sum
|
"a plus b"
a,b terms added or subtracted
commutative: a+b=b+a
associative: (a+b)+c=a+(b+c)
0 is identity: a+0=a
a, -a inverses: a+-a=0 †
|
| subtraction
| a - b
| difference
|
"a minus b" "difference of a and b"
Same as adding the additive inverse: a+-b
Not commutative: a-b≠b-a
Not associative:
(a-b)-c≠a-(b-c)
|
| multiplication
| a × b
a · b
ab (juxtaposition) 5a 5x (5)(6)
a * b
| product
|
"a times b" "the product of a and b"
shortcut repeated addition
a,b factors
commutative: ab=ba
associative: (ab)c=a(bc)
1 is identity: a·1=a
a, 1/a inverses: a·1/a=1 †
distributes over addition: a(b+c)=ab+ac
If ab=0 then either a=0 or b=0
a·0=0
|
| division
| a ÷ b
a / b
| quotient
|
"a divided by b" "quotient of a and b"
shortcut repeated subtraction
Not commutative: a/b≠b/a
Not associative: (a/b)/c≠a/(b/c)
reciprocal of a is 1/a
reciprocal of a/b is b/a
a/b=a(1/b)
|
| exponentiation
| bp (superscript)
b^p
| power
|
"b to the pth power"
b is base, p is exponent or power.
shortcut repeated multiplication
b2 "squared"
b3 "cubed"
b0=1
b1=b
Laws: xmxn=xm+n
(xm)n=xmn
xm/xn=xm-n
x-n=1/xn
x-1=1/x
Logarithm is inverse:
logbx=p means bp=x
NB. x+x=2x x·x=x2
|
|
|
| square root
| √a (radical)
sqrt(a)
| root
| radicand a≥0 √a=a1/2
positive (principal) root
√a2=|a|
√a√a=a
√of non-perfect square integer is irrational
√(ab)=√a√b
|
† number ⊛ inverse = identity
is an agreement on the order of operations so as to not have to use parentheses for every operation:
5+24·3-6/4+1 ≡ (((5+((24)·3))-(6/4))+1)
Addition is the fundamental operation, the others (-,*,/,^) can be defined in terms of it.
Viz. subtraction is addition of additive inverse,
multiplication is repeated addition,
division is repeated subtraction,
exponentiation is repeated multiplication.
Opposites/inverses: addition/subtraction, multiplication/division,
squaring/square-rooting
Z not closed under division. Q not closed under square root.
Distributive law:
