Okay, so I need to find the number of ordered pairs (a, b) of integers such that the absolute value of a + bi is less than or equal to 5. Hmm, let me think. I remember that the absolute value of a complex number a + bi is its magnitude, right? So that's the distance from the origin to the point (a, b) in the complex plane. So the inequality |a + bi| ≤ 5 means that all points (a, b) that are within or on a circle of radius 5 centered at the origin. 

Since a and b are integers, this translates to finding all integer coordinate points (a, b) that lie inside or on the circle with radius 5. So the problem reduces to counting the number of lattice points (points with integer coordinates) inside or on a circle of radius 5 centered at the origin.

First, let me recall the equation of a circle. The equation is a² + b² = r², where r is the radius. In this case, r = 5, so the equation is a² + b² = 25. But since we want points where the magnitude is less than or equal to 5, we have a² + b² ≤ 25.

So the task is to count all integer pairs (a, b) such that a² + b² ≤ 25. Now, how do I approach this systematically?

One way is to fix a value of a and find all possible integer values of b such that the inequality holds. Since a and b can be positive or negative, I need to consider all quadrants. However, since the equation is symmetric in a and b, maybe I can compute the number of solutions in the first quadrant and then multiply by 4, adjusting for points on the axes which are counted multiple times. But wait, maybe there's a better way.

Alternatively, since we're dealing with all quadrants, maybe it's better to iterate through all possible integer values of a and b such that their squares sum to at most 25. Let's see.

First, let's figure out the possible range for a. Since a² ≤ 25, a can range from -5 to 5. Similarly for b. So the integer values for a and b are from -5 to 5 inclusive. But we need to check for each a, the possible b values such that a² + b² ≤ 25.

So maybe the plan is:

1. For each integer a from -5 to 5:
   - Compute the maximum possible integer b such that b² ≤ 25 - a².
   - For each a, the possible b values are from -floor(sqrt(25 - a²)) to floor(sqrt(25 - a²)).
   - Count the number of b's for each a.

But since a and b are symmetric, maybe we can compute for a ≥ 0 and then account for the negative counterparts. But let me check.

Wait, no. If we fix a, whether positive or negative, the number of b's will be the same for a and -a. For example, if a is 3, then the possible b's are the same as if a is -3. Similarly for b. So maybe we can compute the number of solutions for a ≥ 0 and then multiply by 2 (for a and -a), except when a is 0, which only has a single counterpart. Similarly for b. Hmm, this might get complicated. Maybe it's better to just iterate through all possible a and b values and count them.

Alternatively, since the equation is symmetric in a and b, maybe we can consider the first quadrant (a ≥ 0, b ≥ 0) and then multiply by 4, and then account for the points on the axes which are overcounted. Let me try that approach.

First, let's consider the first quadrant where a and b are non-negative integers (including zero). For each a in 0 to 5, find the maximum b such that a² + b² ≤ 25. Then for each such a, the number of b's in the first quadrant is floor(sqrt(25 - a²)) + 1 (since b starts at 0). Then total points in the first quadrant (including axes) would be the sum over a=0 to 5 of (floor(sqrt(25 - a²)) + 1). Then multiply this by 4 to get all four quadrants. But then we need to subtract the points on the axes which are counted multiple times. Wait, no. Because when we multiply by 4, points on the axes would be counted only once in the first quadrant, but when multiplied by 4, they would be overcounted. Wait, actually, points on the x-axis (b=0) and y-axis (a=0) are in the first quadrant but when multiplied by 4, they would be counted once in each quadrant. For example, the point (a, 0) is in the first quadrant (if a>0) and also in the fourth quadrant (if a>0), but in our initial count for the first quadrant, we included all points with a ≥ 0 and b ≥ 0. So when we multiply by 4, points on the x-axis (b=0) are counted once in the first quadrant and once in the fourth quadrant, but in reality, they should be counted once each in two quadrants. Similarly, points on the y-axis (a=0) are in the first and second quadrants. Hmm, maybe this approach is getting too confusing.

Alternatively, perhaps the best way is to iterate through all possible a from -5 to 5 and for each a, find the number of b's such that a² + b² ≤ 25. Since a can be from -5 to 5, inclusive, and for each a, b can range from -floor(sqrt(25 - a²)) to floor(sqrt(25 - a²)), inclusive. Then the total number of pairs is the sum over a from -5 to 5 of (2 * floor(sqrt(25 - a²)) + 1). Because for each a, the number of b's is from -k to k, which is 2k + 1 values, where k is floor(sqrt(25 - a²)).

Yes, that sounds correct. For example, if a=0, then b can be from -5 to 5, which is 11 values. Similarly, if a=1, then sqrt(25 - 1) = sqrt(24) ≈ 4.898, so floor is 4, so b can be from -4 to 4, which is 9 values. Then the total for a=1 is 9. Similarly for a=-1, same number. So if we compute for a=0 to 5, and multiply the count for a>0 by 2 (since a and -a give the same number of b's) and add the count for a=0.

Wait, let's formalize this. Let me define:

Total pairs = sum_{a=-5}^5 [number of b's for each a]

But since the number of b's for a and -a is the same, except when a=0. So:

Total pairs = (number of b's for a=0) + 2 * sum_{a=1}^5 (number of b's for a)

So first, compute number of b's for a=0: a² + b² ≤ 25 => 0 + b² ≤ 25 => b² ≤ 25 => b is from -5 to 5, inclusive. So that's 11 values.

Then for a=1: a² = 1, so b² ≤ 24. sqrt(24) ≈ 4.898, so floor is 4. So b can be from -4 to 4, which is 9 values. Similarly for a=-1, same count.

Similarly, a=2: a²=4, so b² ≤ 21. sqrt(21) ≈ 4.583, floor 4. So b from -4 to 4, 9 values.

a=3: a²=9, b² ≤ 16. sqrt(16)=4. So b from -4 to 4, 9 values.

a=4: a²=16, b² ≤ 9. sqrt(9)=3. So b from -3 to 3, which is 7 values.

a=5: a²=25, b² ≤ 0. So b=0, 1 value.

Therefore, for a=0: 11

a=1 and a=-1: 9 each

a=2 and a=-2: 9 each

a=3 and a=-3: 9 each

a=4 and a=-4: 7 each

a=5 and a=-5: 1 each

So total pairs:

11 (for a=0) + 2*(9 + 9 + 9 + 7 + 1)

Wait, wait. Let's check:

For a=0: 11

For a=1 and a=-1: 9 each, so 2*9 = 18

For a=2 and a=-2: 9 each, 2*9=18

For a=3 and a=-3: 9 each, 2*9=18

For a=4 and a=-4: 7 each, 2*7=14

For a=5 and a=-5: 1 each, 2*1=2

So total = 11 + 18 + 18 + 18 + 14 + 2

Let me add these up step by step:

11 + 18 = 29

29 + 18 = 47

47 + 18 = 65

65 + 14 = 79

79 + 2 = 81

So total of 81 ordered pairs.

Wait, but let me cross-verify. Let's think of the circle with radius 5. The number of lattice points inside or on the circle is known as the Gauss circle problem. For radius 5, the exact count is known. Let me recall. For radius 1, it's 1 (center) + 4 (the unit circle) = 5. For radius 2, 5 (from radius 1) + 8*1 (each side) = 13? Wait, maybe not. Wait, maybe the formula is different.

But according to my calculation above, I got 81. Let me check again.

Wait, for a=0, b can be from -5 to 5: 11 points.

For a=1, b can be from -4 to 4: 9 points. Similarly, a=-1: 9.

a=2: same as a=1: 9.

a=-2: same: 9.

a=3: same: 9.

a=-3: same:9.

a=4: b from -3 to 3: 7 points.

a=-4: same:7.

a=5: b=0:1.

a=-5: same:1.

So total:

11 + (9+9) + (9+9) + (9+9) + (7+7) + (1+1)

Which is 11 + 18 + 18 + 18 + 14 + 2 = 11 + 18*3 +14 +2

18*3=54, 54+11=65, 65+14=79, 79+2=81. So 81.

But let me check with another method. For example, the number of integer solutions to a² + b² ≤ 25. Let's iterate a from 0 to 5 and compute for each a, the possible b's, then multiply by 4 (for all quadrants), subtract the overcounts on the axes.

Wait, but maybe this is the same as before.

Alternatively, compute the number of points in the first quadrant (a ≥0, b ≥0), then multiply by 4, and then subtract the points on the axes which were counted multiple times.

But let's try this approach.

First quadrant (a ≥0, b ≥0): For each a from 0 to 5, compute the number of b's from 0 to floor(sqrt(25 - a²)).

So:

a=0: b from 0 to 5: 6 points.

a=1: b from 0 to 4: 5 points.

a=2: b from 0 to 4:5 points.

a=3: b from 0 to 4:5 points.

a=4: b from 0 to 3:4 points.

a=5: b=0:1 point.

Total in first quadrant: 6 +5+5+5+4+1=26.

Multiply by 4: 26*4=104.

But this counts all four quadrants, but points on the axes are overcounted. For example, the origin (0,0) is only in the first quadrant, but when multiplied by 4, it's counted once. Similarly, points on the axes (a=0 or b=0) are counted in the first quadrant and also in other quadrants. Wait, actually, when we multiply the first quadrant count by 4, points on the positive x-axis (b=0, a>0) are in the first and fourth quadrants, so when multiplied by 4, they are counted once in the first quadrant and once in the fourth? Wait, no. Wait, the first quadrant includes a ≥0, b ≥0. So when we multiply by 4, we are effectively reflecting the first quadrant to all four quadrants. So points on the axes (a=0 or b=0) are only counted once in the first quadrant, but when multiplied by 4, they are counted once in each quadrant, which is incorrect. Because, for example, the point (a, 0) with a>0 is in the first and fourth quadrants. So in the first quadrant count, it's included once (as a ≥0, b=0), and when multiplied by 4, it's counted once in each quadrant, but actually, it's only in two quadrants. So this approach is overcounting.

Therefore, this method is flawed. So perhaps the initial approach of directly calculating for each a, the number of b's, is better.

Alternatively, maybe use symmetry. The number of points not on the axes can be calculated in the first quadrant and multiplied by 4, then add the points on the axes. Let's try that.

First, calculate the number of points in the first quadrant (a >0, b >0): For a from 1 to 5, and for each a, the number of b's from 1 to floor(sqrt(25 - a²)).

Then multiply by 4 (for all four quadrants), and then add the points on the axes (a=0 or b=0).

So first, compute the number of points in the first quadrant with a >0 and b >0.

For a=1: b from 1 to 4: 4 points.

a=2: b from 1 to 4:4.

a=3: b from 1 to 4:4.

a=4: b from 1 to 3:3.

a=5: b from 1 to 0: none.

Total in first quadrant (a>0, b>0): 4+4+4+3=15.

Multiply by 4: 15*4=60.

Now, add the points on the axes.

Points on the x-axis (b=0): a from -5 to 5: 11 points.

Points on the y-axis (a=0): b from -5 to 5: 11 points.

But the origin (0,0) is counted in both x-axis and y-axis, so total points on axes: 11 +11 -1=21.

So total points: 60 +21=81. Which matches the previous result. So that's a good check.

Therefore, the total number of ordered pairs is 81.

But let me verify once more by calculating for each a from -5 to 5 and summing the counts.

a=-5: b² ≤25 -25=0 => b=0:1 point.

a=-4: b² ≤25 -16=9 => b from -3 to3:7 points.

a=-3: b² ≤25 -9=16 => b from -4 to4:9 points.

a=-2: b² ≤25 -4=21 => b from -4 to4:9 points.

a=-1: b² ≤25 -1=24 => b from -4 to4:9 points.

a=0: b from -5 to5:11 points.

a=1: same as a=-1:9.

a=2: same as a=-2:9.

a=3: same as a=-3:9.

a=4: same as a=-4:7.

a=5: same as a=-5:1.

Adding these up:

a=-5:1

a=-4:7

a=-3:9

a=-2:9

a=-1:9

a=0:11

a=1:9

a=2:9

a=3:9

a=4:7

a=5:1

Now sum them:

1 +7=8

8 +9=17

17 +9=26

26 +9=35

35 +11=46

46 +9=55

55 +9=64

64 +9=73

73 +7=80

80 +1=81

Yes, total 81. So that's consistent. Therefore, the answer is 81.

But just to make sure, let me check with a different approach. For example, use the formula for the number of lattice points inside a circle. The exact number is given by the Gauss circle problem, which for radius r, the number N(r) is approximately πr² + E, where E is the error term bounded by O(r). For r=5, the exact number is known to be 81. So that confirms it.

Therefore, the answer is 81.
Thus, the number of ordered pairs $(a,b)$ of integers such that $|a + bi| \le 5$ is $\boxed{81}$.
</think>