Integers

Inajengwa juu ya

Masharti ya awali ya lazima 4 (hadi chini kabisa)

Utajifunza

• Place integers on the number line and interpret negative numbers as opposites, debts, temperatures below zero or positions left of zero.
• Add, subtract and multiply integers, explaining subtraction as adding the additive inverse.
• Prove from the distributive law that a · 0 = 0, (−a)b = −(ab) and (−a)(−b) = ab.
• Explain why every integer has an additive inverse in ℤ while only 1 and −1 have multiplicative inverses in ℤ.

Fasili

• The integers are the whole numbers 0, 1, 2, … together with their negatives −1, −2, −3, ….
• The additive inverse (opposite) of a is the integer −a with a + (−a) = 0.
• Subtraction is defined by a − b = a + (−b).
• A multiplicative inverse of a is a number b with ab = 1.

Nadharia

• Addition and multiplication of integers are commutative and associative, multiplication distributes over addition, 0 is the additive identity and 1 the multiplicative identity.
• Every integer has exactly one additive inverse.
• a · 0 = 0 and (−a)(−b) = ab for all integers a and b, as consequences of the distributive law.
• The only integers with a multiplicative inverse in ℤ are 1 and −1.
• ℤ has no zero divisors: if ab = 0 then a = 0 or b = 0.

Mawazo potofu ya kawaida

• Reading '−' only as 'negative': −a is positive when a is negative, and '−' can also mean 'subtract' or 'the opposite of'.
• Believing (−a)(−b) = ab is an arbitrary convention; it is forced by the distributive law.
• Thinking −5 > −2 because 5 > 2.
• Assuming every integer has an integer reciprocal.

Inaelekea kwa

Ugunduzi wa hiari

• Algebraic structures: monoids, groups and rings Under addition the integers have an identity (0) and every element has an inverse: they form a group. Under multiplication only 1 and −1 have inverses, so ℤ is a ring but not a field.Njia ya Maajabu →
• Groups (ℤ, +) is the infinite cyclic group, generated by 1.Njia ya Maajabu →
• Rings, ideals and fields (ℤ, +, ×) is the prototype of a commutative ring with identity; enlarging it to ℚ gives every non-zero element a multiplicative inverse.Njia ya Maajabu →
• Clock arithmetic: remainders and modular arithmetic On a clock, adding 12 hours changes nothing: arithmetic with remainders wraps the infinite line of integers into a finite circle.Njia ya Maajabu →
• Congruences a ≡ b (mod n) means that n divides a − b; congruence respects addition and multiplication.Njia ya Maajabu →
• Relations, equivalence relations and partitions Grouping integers by their remainder on division by n splits ℤ into n families, the classes of an equivalence relation.Njia ya Maajabu →
• Constructing ℤ, ℚ and ℝ The integers can be constructed from the natural numbers as equivalence classes of pairs (a, b), thought of as a − b.Njia ya Maajabu →
• ← Addition, subtraction and place value to 100

Njia za kujifunza

• Conventional K–12 progression
• Accelerated secondary progression
• Adult beginner pathway

Usomaji zaidi

• I. M. Gelfand and A. Shen, Algebra (1993)
• Hung-Hsi Wu, Understanding Numbers in Elementary School Mathematics (2011)