Editing Addition (section)

=== Decimal system ===
The prerequisite to addition in the [[decimal]] system is the fluent recall or derivation of the 100 single-digit "addition facts". One could [[memorize]] all the facts by [[rote learning|rote]], but pattern-based strategies are more enlightening and, for most people, more efficient:{{sfnp|Fosnot|Dolk|2001|p=99}}
* ''Commutative property'': Mentioned above, using the pattern <math> a + b = b + a </math> reduces the number of "addition facts" from 100 to 55.
* ''One or two more'': Adding 1 or 2 is a basic task, and it can be accomplished through counting on or, ultimately, [[intuition (knowledge)|intuition]].{{sfnp|Fosnot|Dolk|2001|p=99}}
* ''Zero'': Since zero is the additive identity, adding zero is trivial. Nonetheless, in the teaching of arithmetic, some students are introduced to addition as a process that always increases the addends; [[word problem (mathematics education)|word problems]] may help rationalize the "exception" of zero.{{sfnp|Fosnot|Dolk|2001|p=99}}
* ''Doubles'': Adding a number to itself is related to counting by two and to [[multiplication]]. Doubles facts form a backbone for many related facts, and students find them relatively easy to grasp.{{sfnp|Fosnot|Dolk|2001|p=99}}
* ''Near-doubles'': Sums such as 6 + 7 = 13 can be quickly derived from the doubles fact {{nowrap|1=6 + 6 = 12}} by adding one more, or from {{nowrap|1=7 + 7 = 14}} but subtracting one.{{sfnp|Fosnot|Dolk|2001|p=99}}
* ''Five and ten'': Sums of the form 5 + {{mvar|x}} and 10 + {{mvar|x}} are usually memorized early and can be used for deriving other facts. For example, {{nowrap|1=6 + 7 = 13}} can be derived from {{nowrap|1=5 + 7 = 12}} by adding one more.{{sfnp|Fosnot|Dolk|2001|p=99}}
* ''Making ten'': An advanced strategy uses 10 as an intermediate for sums involving 8 or 9; for example, {{nowrap|1=8 + 6 = 8 + 2 + 4 =}} {{nowrap|1=10 + 4 = 14}}.{{sfnp|Fosnot|Dolk|2001|p=99}}
As students grow older, they commit more facts to memory and learn to derive other facts rapidly and fluently. Many students never commit all the facts to memory, but can still find any basic fact quickly.<ref name=Henry/>

==== Carry ====
{{main|Carry (arithmetic)}}
[[File:Addition with carry.png|thumb|upright=1|An addition with [[Carry (mathematics)|carry]]]]
The standard algorithm for adding multidigit numbers is to align the addends vertically and add the columns, starting from the ones column on the right. If a column exceeds nine, the extra digit is "[[carry (arithmetic)|carried]]" into the next column. For example, in the following image, the ones in the addition of {{nowrap|59 + 27}} is 9 + 7 = 16, and the digit 1 is the carry.<ref group=lower-alpha>Some authors think that "carry" may be inappropriate for education; {{harvtxt|van de Walle|2004}}, p. 211 calls it "obsolete and conceptually misleading", preferring the word "trade". However, "carry" remains the standard term.</ref> An alternate strategy starts adding from the most significant digit on the left; this route makes carrying a little clumsier, but it is faster at getting a rough estimate of the sum. There are many alternative methods.

==== Decimal fractions ====
[[Decimal fractions]] can be added by a simple modification of the above process.<ref>Rebecca Wingard-Nelson (2014) ''Decimals and Fractions: It's Easy'' Enslow Publishers, Inc.</ref> One aligns two decimal fractions above each other, with the decimal point in the same location. If necessary, one can add trailing zeros to a shorter decimal to make it the same length as the longer decimal. Finally, one performs the same addition process as above, except the decimal point is placed in the answer, exactly where it was placed in the summands.

As an example, 45.1 + 4.34 can be solved as follows:
    4 5 . 1 0
 +  0 4 . 3 4
 ————————————
    4 9 . 4 4

==== Scientific notation ====
{{main|Scientific notation#Basic operations}}
In [[scientific notation]], numbers are written in the form <math>x=a\times10^{b}</math>, where <math>a</math> is the significand and <math>10^{b}</math> is the exponential part. Addition requires two numbers in scientific notation to be represented using the same exponential part, so that the two significands can simply be added.

For example:
: <math>\begin{align}
&2.34\times10^{-5} + 5.67\times10^{-6} \\
&\quad = 2.34\times10^{-5} + 0.567\times10^{-5} \\
&\quad = 2.907\times10^{-5}.
\end{align}</math>