Rational Number Foundations

No

q=0, so the denominator is zero and no ordinary rational value is defined.

Yes

q=2≠0, so -11/2 is a valid representation.

Yes

q=3≠0, so -10/3 is a valid representation.

Yes

q=4≠0, so -9/4 is a valid representation.

Yes

q=5≠0, so -8/5 is a valid representation.

No

q=0, so the denominator is zero and no ordinary rational value is defined.

Yes

q=7≠0, so -6/7 is a valid representation.

Yes

q=1≠0, so -5/1 is a valid representation.

Yes

q=2≠0, so -4/2 is a valid representation.

Yes

q=3≠0, so -3/3 is a valid representation.

No

q=0, so the denominator is zero and no ordinary rational value is defined.

Yes

q=5≠0, so -1/5 is a valid representation.

Yes

q=6≠0, so 0/6 is a valid representation.

Yes

q=7≠0, so 1/7 is a valid representation.

Yes

q=1≠0, so 2/1 is a valid representation.

No

q=0, so the denominator is zero and no ordinary rational value is defined.

Yes

q=3≠0, so 4/3 is a valid representation.

Yes

q=4≠0, so 5/4 is a valid representation.

Yes

q=5≠0, so 6/5 is a valid representation.

Yes

q=6≠0, so 7/6 is a valid representation.

No

q=0, so the denominator is zero and no ordinary rational value is defined.

Yes

q=1≠0, so 9/1 is a valid representation.

Yes

q=2≠0, so 10/2 is a valid representation.

Yes

q=3≠0, so 11/3 is a valid representation.

Yes

q=4≠0, so 12/4 is a valid representation.

12/1

Because q=-1 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

11/2

Because q=-2 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

10/3

Because q=-3 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

9/4

Because q=-4 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

8/5

Because q=-5 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

7/6

Because q=-6 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

6/7

Because q=-7 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

5/8

Because q=-8 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

4/9

Because q=-9 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

3/1

Because q=-1 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

2/2

Because q=-2 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

1/3

Because q=-3 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

0/4

Because q=-4 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

-1/5

Because q=-5 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

-2/6

Because q=-6 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

-3/7

Because q=-7 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

-4/8

Because q=-8 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

-5/9

Because q=-9 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

-6/1

Because q=-1 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

-7/2

Because q=-2 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

-8/3

Because q=-3 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

-9/4

Because q=-4 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

-10/5

Because q=-5 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

-11/6

Because q=-6 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

-12/7

Because q=-7 is negative, move only the sign to the numerator. The value is unchanged; no gcd reduction is performed.

Yes

-11×2 = -22×1; the fractions are equivalent. This uses exact cross products rather than decimal appearance.

No

-10×4 ≠ -19×2; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.

Yes

-9×6 = -18×3; the fractions are equivalent. This uses exact cross products rather than decimal appearance.

No

-8×8 ≠ -15×4; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.

Yes

-7×10 = -14×5; the fractions are equivalent. This uses exact cross products rather than decimal appearance.

No

-6×12 ≠ -11×6; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.

Yes

-5×14 = -10×7; the fractions are equivalent. This uses exact cross products rather than decimal appearance.

No

-4×2 ≠ -7×1; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.

Yes

-3×4 = -6×2; the fractions are equivalent. This uses exact cross products rather than decimal appearance.

No

-2×6 ≠ -3×3; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.

Yes

-1×8 = -2×4; the fractions are equivalent. This uses exact cross products rather than decimal appearance.

No

0×10 ≠ 1×5; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.

Yes

1×12 = 2×6; the fractions are equivalent. This uses exact cross products rather than decimal appearance.

No

2×14 ≠ 5×7; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.

Yes

3×2 = 6×1; the fractions are equivalent. This uses exact cross products rather than decimal appearance.

No

4×4 ≠ 9×2; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.

Yes

5×6 = 10×3; the fractions are equivalent. This uses exact cross products rather than decimal appearance.

No

6×8 ≠ 13×4; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.

Yes

7×10 = 14×5; the fractions are equivalent. This uses exact cross products rather than decimal appearance.

No

8×12 ≠ 17×6; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.

Yes

9×14 = 18×7; the fractions are equivalent. This uses exact cross products rather than decimal appearance.

No

10×2 ≠ 21×1; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.

Yes

11×4 = 22×2; the fractions are equivalent. This uses exact cross products rather than decimal appearance.

No

12×6 ≠ 25×3; the fractions are not equivalent. This uses exact cross products rather than decimal appearance.

Yes

13×8 = 26×4; the fractions are equivalent. This uses exact cross products rather than decimal appearance.

0/1

The canonical reduced form of zero is 0/1.

-3/2

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -3/2.

9/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 9/4.

-3/1

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -3/1.

15/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 15/4.

-9/2

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -9/2.

21/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 21/4.

-6/1

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -6/1.

27/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 27/4.

-15/2

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -15/2.

33/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 33/4.

-9/1

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -9/1.

39/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 39/4.

-21/2

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -21/2.

45/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 45/4.

-12/1

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -12/1.

51/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 51/4.

-27/2

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -27/2.

57/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 57/4.

-15/1

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -15/1.

63/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 63/4.

-33/2

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -33/2.

69/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 69/4.

-18/1

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is -18/1.

75/4

Divide numerator and denominator by their gcd and normalize the denominator sign; the result is 75/4.