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Real Numbers

Real Numbers. 1. Definition. 2. Properties. 3. Examples. Definition. Real Numbers include: Integers -3,-2,-1,0,1,2,3 Rational Numbers Decimals that can be represented in fraction form that are either terminating or non-terminating and repeating 5/4 = 1.25 177/55 = 3.2181818…

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Real Numbers

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  1. Real Numbers 1 Definition 2 Properties 3 Examples www.themegallery.com

  2. Definition • Real Numbers include: • Integers • -3,-2,-1,0,1,2,3 • Rational Numbers • Decimals that can be represented in fraction form that are either terminating or non-terminating and repeating • 5/4 = 1.25 • 177/55 = 3.2181818… • 1/3 = .33333… • Irrational Numbers • Non-terminating and non-repeating decimals • Π = 3.14159…, √2 = 1.41421…

  3. Properties • Addition is commutative • a + b = b + a • Order does not matter • Addition is associative • a + (b + c) = (a +b) + c • Grouping does not matter • 0 is the additive identity • a + 0 = a • Adding 0 yields the same number www.themegallery.com

  4. Properties (Cont.) • -a is the additive inverse (negative) of a • a + (-a) = 0, 12+(-12)=0 • Adding a number and it’s inverse gives 0 • Multiplication is commutative • ab = ba, 3*4=4*3=12 • Order of multiplication does not change the result • 1 is the multiplicative identity • a * 1 = a • Multiplying 1 yields the same number www.themegallery.com

  5. Properties (Cont.) • If a ≠ 0, 1/a is the multiplicative inverse (reciprocal) of a • a(1/a) = 1, 3(1/3)=1 • Multiplying a non-zero number by its reciprocal yields 1 • Multiplication is distributive over addition • a(b + c) = ab + ac • (a + b)c = ac + bc • Multiplying a number and a sum of two numbers is the same as multiplying each of the two numbers by the multiplier and then adding the products www.themegallery.com

  6. Properties (Cont.) • Trichotomy Law • If a and b are real numbers, then exactly one of the following is true: a=b, a<b, a>b • Definition of Absolute Value • If a ≥ 0, then |a|=a • If a <0, then |a|=-(a) • Distance on a number line • d(A, B) = |B-A| • Law of the signs • If a and b both have the same sign, then ab and a/b are positive • If a and b have different signs, then ab and a/b are negative

  7. Examples • If p, q, r, and s denote real numbers, show that (p+q)(r+s)=pr+ps+qr+qs • (p+q)(r+s) =p(r+s)+q(r+s) =(pr+ps)+(qr+qs) =pr+ps+qr+qs • If x>0, and y<0, determine the sign of x/y + y/x • Since only y is negative, both x\y and y/x will be negative numbers • A negative number increased by another negative number will yield a “more” negative number • If x<1, rewrite |x-1| without using the absolute value symbol • If x<1, then x-1<0 (negative) • By part 2 of the definition of absolute value, |x-1|=-(x-1)=-x+1 or 1-x

  8. Examples • Let A, B, C, and D have coordinates -5, -3, 1, and 6 respectively. Find d(B,D)/\. • d(B,D) = d(-3,6) =|6-(-3)| =|6+3| =|9| =9 www.themegallery.com

  9. Guided Practice • Do Problems on page 16, 1-40 www.themegallery.com

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